Theorem ralxfrALT 3840

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 1884	. 2 |- _V = _V
2		ralxfr.1	. . . 4 |- (y e. B -> A e. B)
3	2	adantl 424	. . 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 424	. . 3 |- ((_V = _V /\ x e. B) -> E.y e. B x = A)
6		ralxfr.3	. . . 4 |- (x = A -> (ph <-> ps))
7	6	adantl 424	. . 3 |- ((_V = _V /\ x = A) -> (ph <-> ps))
8	3, 5, 7	ralxfrd 3837	. 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 163   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294

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