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Theorem ax5eq 32416
Description: Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-5 1748 considered as a metatheorem. Do not use it for later proofs - use ax-5 1748 instead, to avoid reference to the redundant axiom ax-c16 32377.) (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax5eq  |-  ( x  =  y  ->  A. z  x  =  y )
Distinct variable groups:    x, z    y, z

Proof of Theorem ax5eq
StepHypRef Expression
1 ax-c9 32375 . 2  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
2 ax-c16 32377 . 2  |-  ( A. z  z  =  x  ->  ( x  =  y  ->  A. z  x  =  y ) )
3 ax-c16 32377 . 2  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y ) )
41, 2, 3pm2.61ii 168 1  |-  ( x  =  y  ->  A. z  x  =  y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-c9 32375  ax-c16 32377
This theorem is referenced by:  dveeq1-o16  32420
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