Theorem rexxfr2d 4883

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		ralxfr2d.2	. . . 4 |- ( ph -> ( x e. B <-> E. y e. C x = A ) )
3		ralxfr2d.3	. . . . 5 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
4	3	notbid 308	. . . 4 |- ( ( ph /\ x = A ) -> ( -. ps <-> -. ch ) )
5	1, 2, 4	ralxfr2d 4882	. . 3 |- ( ph -> ( A. x e. B -. ps <-> A. y e. C -. ch ) )
6	5	notbid 308	. 2 |- ( ph -> ( -. A. x e. B -. ps <-> -. A. y e. C -. ch ) )
7		dfrex2 2996	. 2 |- ( E. x e. B ps <-> -. A. x e. B -. ps )
8		dfrex2 2996	. 2 |- ( E. y e. C ch <-> -. A. y e. C -. ch )
9	6, 7, 8	3bitr4g 303	1 |- ( ph -> ( E. x e. B ps <-> E. y e. C ch ) )

Syntax hints:   -. wn 3   -> 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:  rexrn  6361  rexima  6497  cnpresti  21092  cnprest  21093  1stcrest  21256  subislly  21284  txrest  21434  trfil2  21691  met1stc  22326  metucn  22376  xrlimcnp  24695  esumlub  30122  esumfsup  30132  ptrest  33408  djhcvat42  36704