Theorem ralxfr2d 4882

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 3215	. . . 4 |- ( A e. V -> E. x x = A )
3	1, 2	syl 17	. . 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 238	. . . . . . 7 |- ( ph -> ( E. y e. C x = A -> x e. B ) )
6		r19.23v 3023	. . . . . . 7 |- ( A. y e. C ( x = A -> x e. B ) <-> ( E. y e. C x = A -> x e. B ) )
7	5, 6	sylibr 224	. . . . . 6 |- ( ph -> A. y e. C ( x = A -> x e. B ) )
8	7	r19.21bi 2932	. . . . 5 |- ( ( ph /\ y e. C ) -> ( x = A -> x e. B ) )
9		eleq1 2689	. . . . 5 |- ( x = A -> ( x e. B <-> A e. B ) )
10	8, 9	mpbidi 231	. . . 4 |- ( ( ph /\ y e. C ) -> ( x = A -> A e. B ) )
11	10	exlimdv 1861	. . 3 |- ( ( ph /\ y e. C ) -> ( E. x x = A -> A e. B ) )
12	3, 11	mpd 15	. 2 |- ( ( ph /\ y e. C ) -> A e. B )
13	4	biimpa 501	. 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 4879	1 |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   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:  rexxfr2d  4883  ralrn  6362  ralima  6498  cnrest2  21090  cnprest2  21094  connsuba  21223  subislly  21284  trfbas2  21647  trfil2  21691  flimrest  21787  fclsrest  21828  tsmssubm  21946  metucn  22376  extoimad  38464