Theorem ceqsalg 2628

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
ceqsalg.1	|- F/ x ps
ceqsalg.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
ceqsalg	|- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)   V( x)

Proof of Theorem ceqsalg

Step	Hyp	Ref	Expression
1		elisset 2614	. . 3 |- ( A e. V -> E. x x = A )
2		nfa1 1475	. . . 4 |- F/ x A. x ( x = A -> ph )
3		ceqsalg.1	. . . 4 |- F/ x ps
4		ceqsalg.2	. . . . . . 7 |- ( x = A -> ( ph <-> ps ) )
5	4	biimpd 142	. . . . . 6 |- ( x = A -> ( ph -> ps ) )
6	5	a2i 11	. . . . 5 |- ( ( x = A -> ph ) -> ( x = A -> ps ) )
7	6	sps 1471	. . . 4 |- ( A. x ( x = A -> ph ) -> ( x = A -> ps ) )
8	2, 3, 7	exlimd 1529	. . 3 |- ( A. x ( x = A -> ph ) -> ( E. x x = A -> ps ) )
9	1, 8	syl5com 29	. 2 |- ( A e. V -> ( A. x ( x = A -> ph ) -> ps ) )
10	4	biimprcd 158	. . 3 |- ( ps -> ( x = A -> ph ) )
11	3, 10	alrimi 1456	. 2 |- ( ps -> A. x ( x = A -> ph ) )
12	9, 11	impbid1 140	1 |- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1283   = wceq 1285   F/wnf 1390   E.wex 1422   e. wcel 1434

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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-v 2604

This theorem is referenced by:  ceqsal  2629  sbc6g  2840  uniiunlem  3083  sucprcreg  4300  funimass4  5256  ralrnmpt2  5646