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Theorem equs5a 1828
Description: A property related to substitution that unlike equs5 1936 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 1756 . 2  |-  F/ x A. x ( x  =  y  ->  ph )
2 ax-11 1715 . . 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 1801 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 1527   E.wex 1528
This theorem is referenced by:  sb4a  1864  equs45f  1929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
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