Theorem ceqsalg 1800

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:   x,A

Proof of Theorem ceqsalg

Step	Hyp	Ref	Expression
1		ceqsalg.2	. . . . . . 7 |- (x = A -> (ph <-> ps))
2	1	biimpd 153	. . . . . 6 |- (x = A -> (ph -> ps))
3	2	a2i 9	. . . . 5 |- ((x = A -> ph) -> (x = A -> ps))
4	3	19.20i 968	. . . 4 |- (A.x(x = A -> ph) -> A.x(x = A -> ps))
5		ceqsalg.1	. . . . 5 |- (ps -> A.xps)
6	5	19.23 1039	. . . 4 |- (A.x(x = A -> ps) <-> (E.x x = A -> ps))
7	4, 6	sylib 198	. . 3 |- (A.x(x = A -> ph) -> (E.x x = A -> ps))
8		elex 1794	. . 3 |- (A e. B -> E.x x = A)
9	7, 8	syl5com 52	. 2 |- (A e. B -> (A.x(x = A -> ph) -> ps))
10	1	biimprcd 156	. . 3 |- (ps -> (x = A -> ph))
11	5, 10	19.21ai 974	. 2 |- (ps -> A.x(x = A -> ph))
12	9, 11	impbid1 515	1 |- (A e. B -> (A.x(x = A -> ph) <-> ps))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 950  E.wex 956   = wceq 1099   e. wcel 1105

This theorem is referenced by:  ceqsal 1801  sbc6g 1926  sucprcreg 4524

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-gen 955  ax-9 1102  ax-12 1104  ax-17 1190  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787

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