Theorem ralxfrd 4879

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.) (Proof shortened by JJ, 7-Aug-2021.)

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 751	. . . 4 |- ( ( ( ph /\ y e. C ) /\ x = A ) -> ( ps <-> ch ) )
4	1, 3	rspcdv 3312	. . 3 |- ( ( ph /\ y e. C ) -> ( A. x e. B ps -> ch ) )
5	4	ralrimdva 2969	. 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 3072	. . . . . 6 |- ( ( A. y e. C ch /\ E. y e. C x = A ) -> E. y e. C ( ch /\ x = A ) )
8	2	exbiri 652	. . . . . . . . 9 |- ( ph -> ( x = A -> ( ch -> ps ) ) )
9	8	com23 86	. . . . . . . 8 |- ( ph -> ( ch -> ( x = A -> ps ) ) )
10	9	impd 447	. . . . . . 7 |- ( ph -> ( ( ch /\ x = A ) -> ps ) )
11	10	rexlimdvw 3034	. . . . . 6 |- ( ph -> ( E. y e. C ( ch /\ x = A ) -> ps ) )
12	7, 11	syl5 34	. . . . 5 |- ( ph -> ( ( A. y e. C ch /\ E. y e. C x = A ) -> ps ) )
13	12	adantr 481	. . . 4 |- ( ( ph /\ x e. B ) -> ( ( A. y e. C ch /\ E. y e. C x = A ) -> ps ) )
14	6, 13	mpan2d 710	. . 3 |- ( ( ph /\ x e. B ) -> ( A. y e. C ch -> ps ) )
15	14	ralrimdva 2969	. 2 |- ( ph -> ( A. y e. C ch -> A. x e. B ps ) )
16	5, 15	impbid 202	1 |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   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:  rexxfrd  4881  ralxfr2d  4882  ralxfr  4886  islindf4  20177  cmpfi  21211  rlimcnp  24692  ispisys2  30216  glbconN  34663  mapdordlem2  36926