Theorem ralxfrALT 4199

Description: Transfer universal quantification from a variable x to another variable y contained in expression A. This proof does not use ralxfrd 4194. (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 2652	. . . . 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 27	. . 3 |- ( A. x e. B ph -> ( y e. C -> ps ) )
6	5	ralrimiv 2391	. 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 2355	. . . . 5 |- F/ y A. y e. C ps
9		nfv 1421	. . . . 5 |- F/ y ph
10		rsp 2369	. . . . . 6 |- ( A. y e. C ps -> ( y e. C -> ps ) )
11	2	biimprcd 149	. . . . . 6 |- ( ps -> ( x = A -> ph ) )
12	10, 11	syl6 29	. . . . 5 |- ( A. y e. C ps -> ( y e. C -> ( x = A -> ph ) ) )
13	8, 9, 12	rexlimd 2430	. . . 4 |- ( A. y e. C ps -> ( E. y e. C x = A -> ph ) )
14	7, 13	syl5 28	. . 3 |- ( A. y e. C ps -> ( x e. B -> ph ) )
15	14	ralrimiv 2391	. 2 |- ( A. y e. C ps -> A. x e. B ph )
16	6, 15	impbii 117	1 |- ( A. x e. B ph <-> A. y e. C ps )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   A.wral 2306   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559

This theorem is referenced by: (None)