Theorem ralbidar 16422

Description: More general form of ralbida 2117.

Hypotheses

Ref	Expression
ralbidar.1	|- (ph -> A.x e. A ph)
ralbidar.2	|- ((ph /\ x e. A) -> (ps <-> ch))

Assertion

Ref	Expression
ralbidar	|- (ph -> (A.x e. A ps <-> A.x e. A ch))

Proof of Theorem ralbidar

Step	Hyp	Ref	Expression
1		ralbidar.1	. . . . 5 |- (ph -> A.x e. A ph)
2		ralbidar.2	. . . . . . 7 |- ((ph /\ x e. A) -> (ps <-> ch))
3	2	ex 402	. . . . . 6 |- (ph -> (x e. A -> (ps <-> ch)))
4	3	ralimi 2168	. . . . 5 |- (A.x e. A ph -> A.x e. A (x e. A -> (ps <-> ch)))
5	1, 4	syl 12	. . . 4 |- (ph -> A.x e. A (x e. A -> (ps <-> ch)))
6		df-ral 2109	. . . 4 |- (A.x e. A (x e. A -> (ps <-> ch)) <-> A.x(x e. A -> (x e. A -> (ps <-> ch))))
7	5, 6	sylib 215	. . 3 |- (ph -> A.x(x e. A -> (x e. A -> (ps <-> ch))))
8		pm2.43 77	. . . . 5 |- ((x e. A -> (x e. A -> (ps <-> ch))) -> (x e. A -> (ps <-> ch)))
9	8	pm5.74d 645	. . . 4 |- ((x e. A -> (x e. A -> (ps <-> ch))) -> ((x e. A -> ps) <-> (x e. A -> ch)))
10	9	alimi 1338	. . 3 |- (A.x(x e. A -> (x e. A -> (ps <-> ch))) -> A.x((x e. A -> ps) <-> (x e. A -> ch)))
11		albi 1344	. . 3 |- (A.x((x e. A -> ps) <-> (x e. A -> ch)) -> (A.x(x e. A -> ps) <-> A.x(x e. A -> ch)))
12	7, 10, 11	3syl 24	. 2 |- (ph -> (A.x(x e. A -> ps) <-> A.x(x e. A -> ch)))
13		df-ral 2109	. 2 |- (A.x e. A ps <-> A.x(x e. A -> ps))
14		df-ral 2109	. 2 |- (A.x e. A ch <-> A.x(x e. A -> ch))
15	12, 13, 14	3bitr4g 614	1 |- (ph -> (A.x e. A ps <-> A.x e. A ch))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   e. wcel 1300  A.wral 2105

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-4 1319  ax-5o 1321

This theorem depends on definitions:  df-bi 164  df-an 242  df-ral 2109

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