Theorem equs4 1610

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 1583	. . 3 |- E.x x = y
2		19.29 1508	. . 3 |- ((A.x(x = y -> ph) /\ E.x x = y) -> E.x((x = y -> ph) /\ x = y))
3	1, 2	mpan2 401	. 2 |- (A.x(x = y -> ph) -> E.x((x = y -> ph) /\ x = y))
4		ancl 301	. . . 4 |- ((x = y -> ph) -> (x = y -> (x = y /\ ph)))
5	4	imp 115	. . 3 |- (((x = y -> ph) /\ x = y) -> (x = y /\ ph))
6	5	eximi 1488	. 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 97  A.wal 1240   = wceq 1242  E.wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  sb2  1647  equs45f  1680