Theorem ralxfr 2905

Description: Transfer universal quantification from a variable x to another variable y contained in expression A.

Hypotheses

Ref	Expression
ralxfr.1	|- (y e. B -> A e. B)
ralxfr.2	|- (x e. B -> E.y e. B x = A)
ralxfr.3	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
ralxfr	|- (A.x e. B ph <-> A.y e. B ps)

Distinct variable groups:   ps,x   ph,y   x,A   x,y,B

Proof of Theorem ralxfr

Step	Hyp	Ref	Expression
1		ralxfr.1	. . . . 5 |- (y e. B -> A e. B)
2		ralxfr.3	. . . . . 6 |- (x = A -> (ph <-> ps))
3	2	rcla4v 1876	. . . . 5 |- (A e. B -> (A.x e. B ph -> ps))
4	1, 3	syl 10	. . . 4 |- (y e. B -> (A.x e. B ph -> ps))
5	4	com12 11	. . 3 |- (A.x e. B ph -> (y e. B -> ps))
6	5	r19.21aiv 1716	. 2 |- (A.x e. B ph -> A.y e. B ps)
7		hbra1 1690	. . . . 5 |- (A.y e. B ps -> A.yA.y e. B ps)
8		ax-17 973	. . . . 5 |- (ph -> A.yph)
9		ra4 1697	. . . . . 6 |- (A.y e. B ps -> (y e. B -> ps))
10	2	biimprcd 156	. . . . . 6 |- (ps -> (x = A -> ph))
11	9, 10	syl6 22	. . . . 5 |- (A.y e. B ps -> (y e. B -> (x = A -> ph)))
12	7, 8, 11	r19.23ad 1748	. . . 4 |- (A.y e. B ps -> (E.y e. B x = A -> ph))
13		ralxfr.2	. . . 4 |- (x e. B -> E.y e. B x = A)
14	12, 13	syl5 21	. . 3 |- (A.y e. B ps -> (x e. B -> ph))
15	14	r19.21aiv 1716	. 2 |- (A.y e. B ps -> A.x e. B ph)
16	6, 15	impbi 157	1 |- (A.x e. B ph <-> A.y e. B ps)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  A.wral 1648  E.wrex 1649

This theorem is referenced by:  rexxfr 2907  infm3 6056  infmsup 6070

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815

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