Another Example -#x(Ax&Bx) 1) -#x(Ax&Bx) A $x(Ax>-Bx) The 2 ways to

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Another Example -#x(Ax&Bx) 1) -#x(Ax&Bx) A $x(Ax>-Bx) The 2 ways to translate ‘No’ 1) -#x(Ax&Bx) A $x(Ax>-Bx) GOAL Quantifier Exchange: -$x = #x- -#x = $x-

Another Example -#x(Ax&Bx) 1) -#x(Ax&Bx) A $x(Ax>-Bx) The 2 ways to translate ‘No’ 1) -#x(Ax&Bx) A 2) -$x(Ax>-Bx) PA ?&-? ?,? &I $x(Ax>-Bx) 2-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Another Example -#x(Ax&Bx) 1) -#x(Ax&Bx) A $x(Ax>-Bx) The 2 ways to translate ‘No’ 1) -#x(Ax&Bx) A 2) -$x(Ax>-Bx) PA ?&-? ?,? &I $x(Ax>-Bx) 2-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Another Example -#x(Ax&Bx) 1) -#x(Ax&Bx) A $x(Ax>-Bx) The 2 ways to translate ‘No’ 1) -#x(Ax&Bx) A 2) -$x(Ax>-Bx) PA 3) $x-(Ax&Bx) 1 QE ?&-? ?,? &I $x(Ax>-Bx) 2-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Another Example -#x(Ax&Bx) 1) -#x(Ax&Bx) A $x(Ax>-Bx) The 2 ways to translate ‘No’ 1) -#x(Ax&Bx) A 2) -$x(Ax>-Bx) PA 3) $x-(Ax&Bx) 1 QE ?&-? ?,? &I $x(Ax>-Bx) 2-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Another Example -#x(Ax&Bx) 1) -#x(Ax&Bx) A $x(Ax>-Bx) The 2 ways to translate ‘No’ 1) -#x(Ax&Bx) A 2) -$x(Ax>-Bx) PA 3) $x-(Ax&Bx) 1 QE 4) #x-(Ax>-Bx) 2 QE ?&-? ?,? &I $x(Ax>-Bx) 2-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Another Example -#x(Ax&Bx) 1) -#x(Ax&Bx) A $x(Ax>-Bx) The 2 ways to translate ‘No’ 1) -#x(Ax&Bx) A 2) -$x(Ax>-Bx) PA 3) $x-(Ax&Bx) 1 QE 4) #x-(Ax>-Bx) 2 QE ?&-? ?,? &I $x(Ax>-Bx) 2-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Another Example -#x(Ax&Bx) 1) -#x(Ax&Bx) A $x(Ax>-Bx) The 2 ways to translate ‘No’ 1) -#x(Ax&Bx) A 2) -$x(Ax>-Bx) PA 3) $x-(Ax&Bx) 1 QE 4) #x-(Ax>-Bx) 2 QE 5) -(Aa>-Ba) 4 #O ?&-? ?,? &I $x(Ax>-Bx) 2-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Another Example -#x(Ax&Bx) 1) -#x(Ax&Bx) A $x(Ax>-Bx) The 2 ways to translate ‘No’ 1) -#x(Ax&Bx) A 2) -$x(Ax>-Bx) PA 3) $x-(Ax&Bx) 1 QE 4) #x-(Ax>-Bx) 2 QE 5) -(Aa>-Ba) 4 #O 6) -(Aa&Ba) 3 $O ?&-? ?,? &I $x(Ax>-Bx) 2-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Another Example -#x(Ax&Bx) 1) -#x(Ax&Bx) A $x(Ax>-Bx) The 2 ways to translate ‘No’ 1) -#x(Ax&Bx) A 2) -$x(Ax>-Bx) PA 3) $x-(Ax&Bx) 1 QE 4) #x-(Ax>-Bx) 2 QE 5) -(Aa>-Ba) 4 #O 6) -(Aa&Ba) 3 $O 7) Aa&Ba 5 AR ?&-? ?,? &I $x(Ax>-Bx) 2-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Another Example -#x(Ax&Bx) 1) -#x(Ax&Bx) A $x(Ax>-Bx) The 2 ways to translate ‘No’ 1) -#x(Ax&Bx) A 2) -$x(Ax>-Bx) PA 3) $x-(Ax&Bx) 1 QE 4) #x-(Ax>-Bx) 2 QE 5) -(Aa>-Ba) 4 #O 6) -(Aa&Ba) 3 $O 7) Aa&Ba 5 AR 8) (Aa&Ba)&-(Aa&Ba) 7,6 &I 9) $x(Ax>-Bx) 2-8 -O Quantifier Exchange: -$x = #x- -#x = $x-