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Theorem equs5a 1912
Description: A property related to substitution that unlike equs5 1944 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 1719 . 2  |-  F/ x A. x ( x  =  y  ->  ph )
2 ax-11 1624 . . 3  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
32imp 420 . 2  |-  ( ( x  =  y  /\  A. y ph )  ->  A. x ( x  =  y  ->  ph ) )
41, 3exlimi 1781 1  |-  ( E. x ( x  =  y  /\  A. y ph )  ->  A. x
( x  =  y  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   A.wal 1532   E.wex 1537    = wceq 1619
This theorem is referenced by:  sb4a  1914  equs45f  1915
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-11 1624  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540
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