Theorem equs5e 1788

Description: A property related to substitution that unlike equs5 1822 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 1583	. . . . 5 |- ( ph -> E. y ph )
2		hbe1 1488	. . . . 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 1593	. 2 |- ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ A. y E. y ph ) )
6		equs5a 1787	. 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 1346   = wceq 1348   E.wex 1485

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-11 1499  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ax11e  1789  sb4e  1798