Theorem ceqsalg 2799

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 2785	. . 3 |- ( A e. V -> E. x x = A )
2		nfa1 1563	. . . 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 144	. . . . . 6 |- ( x = A -> ( ph -> ps ) )
6	5	a2i 11	. . . . 5 |- ( ( x = A -> ph ) -> ( x = A -> ps ) )
7	6	sps 1559	. . . 4 |- ( A. x ( x = A -> ph ) -> ( x = A -> ps ) )
8	2, 3, 7	exlimd 1619	. . 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 160	. . 3 |- ( ps -> ( x = A -> ph ) )
11	3, 10	alrimi 1544	. 2 |- ( ps -> A. x ( x = A -> ph ) )
12	9, 11	impbid1 142	1 |- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1370   = wceq 1372   F/wnf 1482   E.wex 1514   e. wcel 2175

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-v 2773

This theorem is referenced by:  ceqsal  2800  sbc6g  3022  uniiunlem  3281  sucprcreg  4595  funimass4  5623  ralrnmpo  6050