Theorem rexbidar 16423

Description: More general form of rexbida 2118.

Hypotheses

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

Assertion

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

Proof of Theorem rexbidar

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	pm5.32d 709	. . . 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		exbi 1397	. . 3 |- (A.x((x e. A /\ ps) <-> (x e. A /\ ch)) -> (E.x(x e. A /\ ps) <-> E.x(x e. A /\ ch)))
12	7, 10, 11	3syl 24	. 2 |- (ph -> (E.x(x e. A /\ ps) <-> E.x(x e. A /\ ch)))
13		df-rex 2110	. 2 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
14		df-rex 2110	. 2 |- (E.x e. A ch <-> E.x(x e. A /\ ch))
15	12, 13, 14	3bitr4g 614	1 |- (ph -> (E.x e. A ps <-> E.x e. A ch))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   e. wcel 1300  E.wex 1326  A.wral 2105  E.wrex 2106

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-ex 1327  df-ral 2109  df-rex 2110

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