Theorem ralxfrALT 4165

Description: Transfer universal quantification from a variable x to another variable y contained in expression A. This proof does not use ralxfrd 4160. (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 2646	. . . . 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 2385	. 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 2349	. . . . 5 |- F/yA.y e. C ps
9		nfv 1418	. . . . 5 |- F/yph
10		rsp 2363	. . . . . 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 2424	. . . 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 2385	. 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 1242   e. wcel 1390  A.wral 2300  E.wrex 2301

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553

This theorem is referenced by: (None)