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Theorem equs5a 1691
Description: A property related to substitution that unlike equs5 1726 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 1449 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  A. x A. x ( x  =  y  ->  ph ) )
2 ax-11 1413 . . 3  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
32imp 119 . 2  |-  ( ( x  =  y  /\  A. y ph )  ->  A. x ( x  =  y  ->  ph ) )
41, 3exlimih 1500 1  |-  ( E. x ( x  =  y  /\  A. y ph )  ->  A. x
( x  =  y  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   A.wal 1257   E.wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-gen 1354  ax-ie2 1399  ax-11 1413  ax-ial 1443
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  equs5e  1692  sb4a  1698  equs45f  1699
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