Theorem ceqsalg 2884

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/xps
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 2870	. . 3 |- (A e. V -> E.x x = A)
2		nfa1 1788	. . . 4 |- F/xA.x(x = A -> ph)
3		ceqsalg.1	. . . 4 |- F/xps
4		ceqsalg.2	. . . . . . 7 |- (x = A -> (ph <-> ps))
5	4	biimpd 198	. . . . . 6 |- (x = A -> (ph -> ps))
6	5	a2i 12	. . . . 5 |- ((x = A -> ph) -> (x = A -> ps))
7	6	sps 1754	. . . 4 |- (A.x(x = A -> ph) -> (x = A -> ps))
8	2, 3, 7	exlimd 1806	. . 3 |- (A.x(x = A -> ph) -> (E.x x = A -> ps))
9	1, 8	syl5com 26	. 2 |- (A e. V -> (A.x(x = A -> ph) -> ps))
10	4	biimprcd 216	. . 3 |- (ps -> (x = A -> ph))
11	3, 10	alrimi 1765	. 2 |- (ps -> A.x(x = A -> ph))
12	9, 11	impbid1 194	1 |- (A e. V -> (A.x(x = A -> ph) <-> ps))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541   F/wnf 1544   = wceq 1642   e. wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  ceqsal  2885  sbc6g  3072  uniiunlem  3354