Theorem cbvrexf 3166

Description: Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem cbvrexf

Step	Hyp	Ref	Expression
1		cbvralf.1	. . . 4 |- F/_ x A
2		cbvralf.2	. . . 4 |- F/_ y A
3		cbvralf.3	. . . . 5 |- F/ y ph
4	3	nfn 1784	. . . 4 |- F/ y -. ph
5		cbvralf.4	. . . . 5 |- F/ x ps
6	5	nfn 1784	. . . 4 |- F/ x -. ps
7		cbvralf.5	. . . . 5 |- ( x = y -> ( ph <-> ps ) )
8	7	notbid 308	. . . 4 |- ( x = y -> ( -. ph <-> -. ps ) )
9	1, 2, 4, 6, 8	cbvralf 3165	. . 3 |- ( A. x e. A -. ph <-> A. y e. A -. ps )
10	9	notbii 310	. 2 |- ( -. A. x e. A -. ph <-> -. A. y e. A -. ps )
11		dfrex2 2996	. 2 |- ( E. x e. A ph <-> -. A. x e. A -. ph )
12		dfrex2 2996	. 2 |- ( E. y e. A ps <-> -. A. y e. A -. ps )
13	10, 11, 12	3bitr4i 292	1 |- ( E. x e. A ph <-> E. y e. A ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   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-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918

This theorem is referenced by:  cbvrex  3168  reusv2lem4  4872  reusv2  4874  nnwof  11754  cbviunf  29372  ac6sf2  29429  dfimafnf  29436  aciunf1lem  29462  bnj1400  30906  phpreu  33393  poimirlem26  33435  indexa  33528  evth2f  39174  fvelrnbf  39177  evthf  39186  eliin2f  39287  stoweidlem34  40251  ovnlerp  40776