Theorem ceqsalg 2627

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 2613	. . 3 |- ( A e. V -> E. x x = A )
2		nfa1 1474	. . . 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 1470	. . . 4 |- ( A. x ( x = A -> ph ) -> ( x = A -> ps ) )
8	2, 3, 7	exlimd 1528	. . 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 1455	. 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 1282   = wceq 1284   F/wnf 1389   E.wex 1421   e. wcel 1433

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-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by:  ceqsal  2628  sbc6g  2839  uniiunlem  3082  sucprcreg  4292  funimass4  5245  ralrnmpt2  5635