Theorem equs5a 1782

Description: A property related to substitution that unlike equs5 1817 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

Step	Hyp	Ref	Expression
1		hba1 1528	. 2 |- ( A. x ( x = y -> ph ) -> A. x A. x ( x = y -> ph ) )
2		ax-11 1494	. . 3 |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
3	2	imp 123	. 2 |- ( ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )
4	1, 3	exlimih 1581	1 |- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1341   E.wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-gen 1437  ax-ie2 1482  ax-11 1494  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  equs5e  1783  sb4a  1789  equs45f  1790