Theorem rexxfr2d 4450

Description: Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

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
rexxfr2d	|- ( ph -> ( E. x e. B ps <-> E. 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 rexxfr2d

Step	Hyp	Ref	Expression
1		ralxfr2d.1	. . . 4 |- ( ( ph /\ y e. C ) -> A e. V )
2		elisset 2744	. . . 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 157	. . . . . . 7 |- ( ph -> ( E. y e. C x = A -> x e. B ) )
6		r19.23v 2579	. . . . . . 7 |- ( A. y e. C ( x = A -> x e. B ) <-> ( E. y e. C x = A -> x e. B ) )
7	5, 6	sylibr 133	. . . . . 6 |- ( ph -> A. y e. C ( x = A -> x e. B ) )
8	7	r19.21bi 2558	. . . . 5 |- ( ( ph /\ y e. C ) -> ( x = A -> x e. B ) )
9		eleq1 2233	. . . . 5 |- ( x = A -> ( x e. B <-> A e. B ) )
10	8, 9	mpbidi 150	. . . 4 |- ( ( ph /\ y e. C ) -> ( x = A -> A e. B ) )
11	10	exlimdv 1812	. . 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 294	. 2 |- ( ( ph /\ x e. B ) -> E. y e. C x = A )
14		ralxfr2d.3	. 2 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
15	12, 13, 14	rexxfrd 4448	1 |- ( ph -> ( E. x e. B ps <-> E. y e. C ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141   A.wral 2448   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732

This theorem is referenced by:  rexrn  5633  rexima  5734  cnptopresti  13032  cnptoprest  13033  txrest  13070