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Theorem equs5a 1804
Description: A property related to substitution that unlike equs5 1839 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
Assertion
Ref Expression
equs5a  |-  ( E. x ( x  =  y  /\  A. y ph )  ->  A. x
( x  =  y  ->  ph ) )

Proof of Theorem equs5a
StepHypRef Expression
1 hba1 1550 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  A. x A. x ( x  =  y  ->  ph ) )
2 ax-11 1516 . . 3  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
32imp 124 . 2  |-  ( ( x  =  y  /\  A. y ph )  ->  A. x ( x  =  y  ->  ph ) )
41, 3exlimih 1603 1  |-  ( E. x ( x  =  y  /\  A. y ph )  ->  A. x
( x  =  y  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1361   E.wex 1502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-gen 1459  ax-ie2 1504  ax-11 1516  ax-ial 1544
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  equs5e  1805  sb4a  1811  equs45f  1812
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