Theorem ralxfrALT 4452

Description: Transfer universal quantification from a variable x to another variable y contained in expression A. This proof does not use ralxfrd 4447. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

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

Allowed substitution hints:   ph( x)   ps( y)   A( y)   C( y)

Proof of Theorem ralxfrALT

Step	Hyp	Ref	Expression
1		ralxfr.1	. . . . 5 |- ( y e. C -> A e. B )
2		ralxfr.3	. . . . . 6 |- ( x = A -> ( ph <-> ps ) )
3	2	rspcv 2830	. . . . 5 |- ( A e. B -> ( A. x e. B ph -> ps ) )
4	1, 3	syl 14	. . . 4 |- ( y e. C -> ( A. x e. B ph -> ps ) )
5	4	com12 30	. . 3 |- ( A. x e. B ph -> ( y e. C -> ps ) )
6	5	ralrimiv 2542	. 2 |- ( A. x e. B ph -> A. y e. C ps )
7		ralxfr.2	. . . 4 |- ( x e. B -> E. y e. C x = A )
8		nfra1 2501	. . . . 5 |- F/ y A. y e. C ps
9		nfv 1521	. . . . 5 |- F/ y ph
10		rsp 2517	. . . . . 6 |- ( A. y e. C ps -> ( y e. C -> ps ) )
11	2	biimprcd 159	. . . . . 6 |- ( ps -> ( x = A -> ph ) )
12	10, 11	syl6 33	. . . . 5 |- ( A. y e. C ps -> ( y e. C -> ( x = A -> ph ) ) )
13	8, 9, 12	rexlimd 2584	. . . 4 |- ( A. y e. C ps -> ( E. y e. C x = A -> ph ) )
14	7, 13	syl5 32	. . 3 |- ( A. y e. C ps -> ( x e. B -> ph ) )
15	14	ralrimiv 2542	. 2 |- ( A. y e. C ps -> A. x e. B ph )
16	6, 15	impbii 125	1 |- ( A. x e. B ph <-> A. y e. C ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732

This theorem is referenced by: (None)