Theorem ralxfr2d 4458

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 2749	. . . 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 158	. . . . . . 7 |- ( ph -> ( E. y e. C x = A -> x e. B ) )
6		r19.23v 2584	. . . . . . 7 |- ( A. y e. C ( x = A -> x e. B ) <-> ( E. y e. C x = A -> x e. B ) )
7	5, 6	sylibr 134	. . . . . 6 |- ( ph -> A. y e. C ( x = A -> x e. B ) )
8	7	r19.21bi 2563	. . . . 5 |- ( ( ph /\ y e. C ) -> ( x = A -> x e. B ) )
9		eleq1 2238	. . . . 5 |- ( x = A -> ( x e. B <-> A e. B ) )
10	8, 9	mpbidi 151	. . . 4 |- ( ( ph /\ y e. C ) -> ( x = A -> A e. B ) )
11	10	exlimdv 1817	. . 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 296	. 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 4456	1 |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   E.wex 1490   e. wcel 2146   A.wral 2453   E.wrex 2454

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737

This theorem is referenced by:  ralrn  5646  ralima  5747  cnrest2  13305  cnptoprest2  13309