Theorem ralxfrALT 4887

Description: Alternate proof of ralxfr 4886 which does not use ralxfrd 4879. (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 3305	. . . . 5 |- ( A e. B -> ( A. x e. B ph -> ps ) )
4	1, 3	syl 17	. . . 4 |- ( y e. C -> ( A. x e. B ph -> ps ) )
5	4	com12 32	. . 3 |- ( A. x e. B ph -> ( y e. C -> ps ) )
6	5	ralrimiv 2965	. 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 2941	. . . . 5 |- F/ y A. y e. C ps
9		nfv 1843	. . . . 5 |- F/ y ph
10		rsp 2929	. . . . . 6 |- ( A. y e. C ps -> ( y e. C -> ps ) )
11	2	biimprcd 240	. . . . . 6 |- ( ps -> ( x = A -> ph ) )
12	10, 11	syl6 35	. . . . 5 |- ( A. y e. C ps -> ( y e. C -> ( x = A -> ph ) ) )
13	8, 9, 12	rexlimd 3026	. . . 4 |- ( A. y e. C ps -> ( E. y e. C x = A -> ph ) )
14	7, 13	syl5 34	. . 3 |- ( A. y e. C ps -> ( x e. B -> ph ) )
15	14	ralrimiv 2965	. 2 |- ( A. y e. C ps -> A. x e. B ph )
16	6, 15	impbii 199	1 |- ( A. x e. B ph <-> A. y e. C ps )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202

This theorem is referenced by: (None)