Theorem equs4 1749

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 1720	. . 3 |- E. x x = y
2		19.29 1644	. . 3 |- ( ( A. x ( x = y -> ph ) /\ E. x x = y ) -> E. x ( ( x = y -> ph ) /\ x = y ) )
3	1, 2	mpan2 425	. 2 |- ( A. x ( x = y -> ph ) -> E. x ( ( x = y -> ph ) /\ x = y ) )
4		ancl 318	. . . 4 |- ( ( x = y -> ph ) -> ( x = y -> ( x = y /\ ph ) ) )
5	4	imp 124	. . 3 |- ( ( ( x = y -> ph ) /\ x = y ) -> ( x = y /\ ph ) )
6	5	eximi 1624	. 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 104   A.wal 1371   = wceq 1373   E.wex 1516

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  sb2  1791  equs45f  1826