Theorem cbvrmo 2576

Description: Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)

Hypotheses

Ref	Expression
cbvral.1	|- F/ y ph
cbvral.2	|- F/ x ps
cbvral.3	|- ( x = y -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   x, A   y, A

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem cbvrmo

Step	Hyp	Ref	Expression
1		cbvral.1	. . . 4 |- F/ y ph
2		cbvral.2	. . . 4 |- F/ x ps
3		cbvral.3	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvrex 2574	. . 3 |- ( E. x e. A ph <-> E. y e. A ps )
5	1, 2, 3	cbvreu 2575	. . 3 |- ( E! x e. A ph <-> E! y e. A ps )
6	4, 5	imbi12i 237	. 2 |- ( ( E. x e. A ph -> E! x e. A ph ) <-> ( E. y e. A ps -> E! y e. A ps ) )
7		rmo5 2569	. 2 |- ( E* x e. A ph <-> ( E. x e. A ph -> E! x e. A ph ) )
8		rmo5 2569	. 2 |- ( E* y e. A ps <-> ( E. y e. A ps -> E! y e. A ps ) )
9	6, 7, 8	3bitr4i 210	1 |- ( E* x e. A ph <-> E* y e. A ps )

Syntax hints:   -> wi 4   <-> wb 103   F/wnf 1389   E.wrex 2349   E!wreu 2350   E*wrmo 2351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-reu 2355  df-rmo 2356

This theorem is referenced by:  cbvrmov  2580  cbvdisj  3776