Theorem cbvrexcsf 3566

Description: A more general version of cbvrexf 3166 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)

Hypotheses

Ref	Expression
cbvralcsf.1	|- F/_ y A
cbvralcsf.2	|- F/_ x B
cbvralcsf.3	|- F/ y ph
cbvralcsf.4	|- F/ x ps
cbvralcsf.5	|- ( x = y -> A = B )
cbvralcsf.6	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvrexcsf	|- ( E. x e. A ph <-> E. y e. B ps )

Proof of Theorem cbvrexcsf

Step	Hyp	Ref	Expression
1		cbvralcsf.1	. . . 4 |- F/_ y A
2		cbvralcsf.2	. . . 4 |- F/_ x B
3		cbvralcsf.3	. . . . 5 |- F/ y ph
4	3	nfn 1784	. . . 4 |- F/ y -. ph
5		cbvralcsf.4	. . . . 5 |- F/ x ps
6	5	nfn 1784	. . . 4 |- F/ x -. ps
7		cbvralcsf.5	. . . 4 |- ( x = y -> A = B )
8		cbvralcsf.6	. . . . 5 |- ( x = y -> ( ph <-> ps ) )
9	8	notbid 308	. . . 4 |- ( x = y -> ( -. ph <-> -. ps ) )
10	1, 2, 4, 6, 7, 9	cbvralcsf 3565	. . 3 |- ( A. x e. A -. ph <-> A. y e. B -. ps )
11	10	notbii 310	. 2 |- ( -. A. x e. A -. ph <-> -. A. y e. B -. ps )
12		dfrex2 2996	. 2 |- ( E. x e. A ph <-> -. A. x e. A -. ph )
13		dfrex2 2996	. 2 |- ( E. y e. B ps <-> -. A. y e. B -. ps )
14	11, 12, 13	3bitr4i 292	1 |- ( E. x e. A ph <-> E. y e. B ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   F/wnf 1708   F/_wnfc 2751   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-sbc 3436  df-csb 3534

This theorem is referenced by:  cbvrexv2  3570