Theorem equs5a 1675

Description: A property related to substitution that unlike equs5 1710 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 1433	. 2 |- ( A. x ( x = y -> ph ) -> A. x A. x ( x = y -> ph ) )
2		ax-11 1397	. . 3 |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
3	2	imp 115	. 2 |- ( ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )
4	1, 3	exlimih 1484	1 |- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-gen 1338  ax-ie2 1383  ax-11 1397  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  equs5e  1676  sb4a  1682  equs45f  1683