Theorem equs5e 1775

Description: A property related to substitution that unlike equs5 1809 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 1570	. . . . 5 |- ( ph -> E. y ph )
2		hbe1 1475	. . . . 5 |- ( E. y ph -> A. y E. y ph )
3	1, 2	syl 14	. . . 4 |- ( ph -> A. y E. y ph )
4	3	anim2i 340	. . 3 |- ( ( x = y /\ ph ) -> ( x = y /\ A. y E. y ph ) )
5	4	eximi 1580	. 2 |- ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ A. y E. y ph ) )
6		equs5a 1774	. 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 103   A.wal 1333   = wceq 1335   E.wex 1472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-11 1486  ax-4 1490  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ax11e  1776  sb4e  1785