Theorem ralxfr2d 4196

Description: Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by Mario Carneiro, 20-Aug-2014.)

Hypotheses

Ref	Expression
ralxfr2d.1	|- ( ( ph /\ y e. C ) -> A e. V )
ralxfr2d.2	|- ( ph -> ( x e. B <-> E. y e. C x = A ) )
ralxfr2d.3	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
ralxfr2d	|- ( 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)   V( x, y)

Proof of Theorem ralxfr2d

Step	Hyp	Ref	Expression
1		ralxfr2d.1	. . . 4 |- ( ( ph /\ y e. C ) -> A e. V )
2		elisset 2568	. . . 4 |- ( A e. V -> E. x x = A )
3	1, 2	syl 14	. . 3 |- ( ( ph /\ y e. C ) -> E. x x = A )
4		ralxfr2d.2	. . . . . . . 8 |- ( ph -> ( x e. B <-> E. y e. C x = A ) )
5	4	biimprd 147	. . . . . . 7 |- ( ph -> ( E. y e. C x = A -> x e. B ) )
6		r19.23v 2425	. . . . . . 7 |- ( A. y e. C ( x = A -> x e. B ) <-> ( E. y e. C x = A -> x e. B ) )
7	5, 6	sylibr 137	. . . . . 6 |- ( ph -> A. y e. C ( x = A -> x e. B ) )
8	7	r19.21bi 2407	. . . . 5 |- ( ( ph /\ y e. C ) -> ( x = A -> x e. B ) )
9		eleq1 2100	. . . . 5 |- ( x = A -> ( x e. B <-> A e. B ) )
10	8, 9	mpbidi 140	. . . 4 |- ( ( ph /\ y e. C ) -> ( x = A -> A e. B ) )
11	10	exlimdv 1700	. . 3 |- ( ( ph /\ y e. C ) -> ( E. x x = A -> A e. B ) )
12	3, 11	mpd 13	. 2 |- ( ( ph /\ y e. C ) -> A e. B )
13	4	biimpa 280	. 2 |- ( ( ph /\ x e. B ) -> E. y e. C x = A )
14		ralxfr2d.3	. 2 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
15	12, 13, 14	ralxfrd 4194	1 |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   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:  ralrn  5305  ralima  5395