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Theorem ax17eq 2135
Description: Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1606 considered as a metatheorem. Do not use it for later proofs - use ax-17 1606 instead, to avoid reference to the redundant axiom ax-16 2096.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax17eq  |-  ( x  =  y  ->  A. z  x  =  y )
Distinct variable groups:    x, z    y, z

Proof of Theorem ax17eq
StepHypRef Expression
1 ax-12o 2094 . 2  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
2 ax-16 2096 . 2  |-  ( A. z  z  =  x  ->  ( x  =  y  ->  A. z  x  =  y ) )
3 ax-16 2096 . 2  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y ) )
41, 2, 3pm2.61ii 157 1  |-  ( x  =  y  ->  A. z  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530
This theorem is referenced by:  dveeq2-o16  2137  dveeq1-o16  2140
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-12o 2094  ax-16 2096
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