Theorem ralxfrd 4497

Description: Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem ralxfrd

Step	Hyp	Ref	Expression
1		ralxfrd.1	. . . 4 |- ( ( ph /\ y e. C ) -> A e. B )
2		ralxfrd.3	. . . . 5 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
3	2	adantlr 477	. . . 4 |- ( ( ( ph /\ y e. C ) /\ x = A ) -> ( ps <-> ch ) )
4	1, 3	rspcdv 2871	. . 3 |- ( ( ph /\ y e. C ) -> ( A. x e. B ps -> ch ) )
5	4	ralrimdva 2577	. 2 |- ( ph -> ( A. x e. B ps -> A. y e. C ch ) )
6		ralxfrd.2	. . . 4 |- ( ( ph /\ x e. B ) -> E. y e. C x = A )
7		r19.29 2634	. . . . 5 |- ( ( A. y e. C ch /\ E. y e. C x = A ) -> E. y e. C ( ch /\ x = A ) )
8	2	biimprd 158	. . . . . . . . 9 |- ( ( ph /\ x = A ) -> ( ch -> ps ) )
9	8	expimpd 363	. . . . . . . 8 |- ( ph -> ( ( x = A /\ ch ) -> ps ) )
10	9	ancomsd 269	. . . . . . 7 |- ( ph -> ( ( ch /\ x = A ) -> ps ) )
11	10	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ x e. B ) /\ y e. C ) -> ( ( ch /\ x = A ) -> ps ) )
12	11	rexlimdva 2614	. . . . 5 |- ( ( ph /\ x e. B ) -> ( E. y e. C ( ch /\ x = A ) -> ps ) )
13	7, 12	syl5 32	. . . 4 |- ( ( ph /\ x e. B ) -> ( ( A. y e. C ch /\ E. y e. C x = A ) -> ps ) )
14	6, 13	mpan2d 428	. . 3 |- ( ( ph /\ x e. B ) -> ( A. y e. C ch -> ps ) )
15	14	ralrimdva 2577	. 2 |- ( ph -> ( A. y e. C ch -> A. x e. B ps ) )
16	5, 15	impbid 129	1 |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765

This theorem is referenced by:  ralxfr2d  4499  ralxfr  4501