Theorem equs5e 1809

Description: A property related to substitution that unlike equs5 1843 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.)

Assertion

Ref	Expression
equs5e	|- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) )

Proof of Theorem equs5e

Step	Hyp	Ref	Expression
1		19.8a 1604	. . . . 5 |- ( ph -> E. y ph )
2		hbe1 1509	. . . . 5 |- ( E. y ph -> A. y E. y ph )
3	1, 2	syl 14	. . . 4 |- ( ph -> A. y E. y ph )
4	3	anim2i 342	. . 3 |- ( ( x = y /\ ph ) -> ( x = y /\ A. y E. y ph ) )
5	4	eximi 1614	. 2 |- ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ A. y E. y ph ) )
6		equs5a 1808	. 2 |- ( E. x ( x = y /\ A. y E. y ph ) -> A. x ( x = y -> E. y ph ) )
7	5, 6	syl 14	1 |- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   = wceq 1364   E.wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-11 1520  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ax11e  1810  sb4e  1819