Theorem ceqsalg 1362

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis.

Hypotheses

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

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem ceqsalg

Step	Hyp	Ref	Expression
1		ceqsalg.2	. . . . . . . 8 |- (x = A -> (ph <-> ps))
2	1	biimpd 135	. . . . . . 7 |- (x = A -> (ph -> ps))
3	2	a2i 8	. . . . . 6 |- ((x = A -> ph) -> (x = A -> ps))
4	3	19.20i 691	. . . . 5 |- (A.x(x = A -> ph) -> A.x(x = A -> ps))
5		ceqsalg.1	. . . . . 6 |- (ps -> A.xps)
6	5	19.23 745	. . . . 5 |- (A.x(x = A -> ps) <-> (E.x x = A -> ps))
7	4, 6	sylib 173	. . . 4 |- (A.x(x = A -> ph) -> (E.x x = A -> ps))
8		elex 1356	. . . 4 |- (A e. B -> E.x x = A)
9	7, 8	syl5 22	. . 3 |- (A.x(x = A -> ph) -> (A e. B -> ps))
10	9	com12 13	. 2 |- (A e. B -> (A.x(x = A -> ph) -> ps))
11	1	biimprcd 138	. . . 4 |- (ps -> (x = A -> ph))
12	5, 11	19.21ai 740	. . 3 |- (ps -> A.x(x = A -> ph))
13	12	a1i 7	. 2 |- (A e. B -> (ps -> A.x(x = A -> ph)))
14	10, 13	impbid 397	1 |- (A e. B -> (A.x(x = A -> ph) <-> ps))

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672  E.wex 678   = wceq 1091   e. wcel 1092

This theorem is referenced by:  ceqsal 1363  sbc6g 1451  sucprcreg 3451

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

metamath.org