Theorem equs4 1653

Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)

Assertion

Ref	Expression
equs4	|- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )

Proof of Theorem equs4

Step	Hyp	Ref	Expression
1		a9e 1626	. . 3 |- E. x x = y
2		19.29 1551	. . 3 |- ( ( A. x ( x = y -> ph ) /\ E. x x = y ) -> E. x ( ( x = y -> ph ) /\ x = y ) )
3	1, 2	mpan2 415	. 2 |- ( A. x ( x = y -> ph ) -> E. x ( ( x = y -> ph ) /\ x = y ) )
4		ancl 311	. . . 4 |- ( ( x = y -> ph ) -> ( x = y -> ( x = y /\ ph ) ) )
5	4	imp 122	. . 3 |- ( ( ( x = y -> ph ) /\ x = y ) -> ( x = y /\ ph ) )
6	5	eximi 1531	. 2 |- ( E. x ( ( x = y -> ph ) /\ x = y ) -> E. x ( x = y /\ ph ) )
7	3, 6	syl 14	1 |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   = wceq 1284   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  sb2  1690  equs45f  1723