Theorem ralxfrALT 2896

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
ralxfrALT	|- (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 ralxfrALT

Step	Hyp	Ref	Expression
1		eqid 1474	. 2 |- V = V
2		ralxfr.1	. . . 4 |- (y e. B -> A e. B)
3	2	adantl 388	. . 3 |- ((V = V /\ y e. B) -> A e. B)
4		ralxfr.2	. . . 4 |- (x e. B -> E.y e. B x = A)
5	4	adantl 388	. . 3 |- ((V = V /\ x e. B) -> E.y e. B x = A)
6		ralxfr.3	. . . 4 |- (x = A -> (ph <-> ps))
7	6	adantl 388	. . 3 |- ((V = V /\ x = A) -> (ph <-> ps))
8	3, 5, 7	ralxfrd 2893	. 2 |- (V = V -> (A.x e. B ph <-> A.y e. B ps))
9	1, 8	ax-mp 7	1 |- (A.x e. B ph <-> A.y e. B ps)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 955   e. wcel 957  A.wral 1643  E.wrex 1644  Vcvv 1808

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ral 1647  df-rex 1648  df-v 1809

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