Theorem ceqsalgOLD 2315

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
ceqsalgOLD	|- (A e. B -> (A.x(x = A -> ph) <-> ps))

Distinct variable group:   x,A

Proof of Theorem ceqsalgOLD

Step	Hyp	Ref	Expression
1		ceqsalg.2	. . . . . . 7 |- (x = A -> (ph <-> ps))
2	1	biimpd 170	. . . . . 6 |- (x = A -> (ph -> ps))
3	2	a2i 10	. . . . 5 |- ((x = A -> ph) -> (x = A -> ps))
4	3	alimi 1338	. . . 4 |- (A.x(x = A -> ph) -> A.x(x = A -> ps))
5		ceqsalg.1	. . . . 5 |- (ps -> A.xps)
6	5	19.23 1411	. . . 4 |- (A.x(x = A -> ps) <-> (E.x x = A -> ps))
7	4, 6	sylib 215	. . 3 |- (A.x(x = A -> ph) -> (E.x x = A -> ps))
8		elex 2302	. . 3 |- (A e. B -> E.x x = A)
9	7, 8	syl5com 63	. 2 |- (A e. B -> (A.x(x = A -> ph) -> ps))
10	1	biimprcd 173	. . 3 |- (ps -> (x = A -> ph))
11	5, 10	19.21ai 1345	. 2 |- (ps -> A.x(x = A -> ph))
12	9, 11	impbid1 575	1 |- (A e. B -> (A.x(x = A -> ph) <-> ps))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

Copyright terms: Public domain