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Theorem equs5a 1840
Description: A property related to substitution that unlike equs5 1949 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 nfa1 1768 . 2  |-  F/ x A. x ( x  =  y  ->  ph )
2 ax-11 1727 . . 3  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
32imp 418 . 2  |-  ( ( x  =  y  /\  A. y ph )  ->  A. x ( x  =  y  ->  ph ) )
41, 3exlimi 1813 1  |-  ( E. x ( x  =  y  /\  A. y ph )  ->  A. x
( x  =  y  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531
This theorem is referenced by:  sb4a  1876  equs45f  1942  equs45fNEW7  29593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535
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