Theorem cbvrmo 2728

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 2726	. . 3 |- ( E. x e. A ph <-> E. y e. A ps )
5	1, 2, 3	cbvreu 2727	. . 3 |- ( E! x e. A ph <-> E! y e. A ps )
6	4, 5	imbi12i 239	. 2 |- ( ( E. x e. A ph -> E! x e. A ph ) <-> ( E. y e. A ps -> E! y e. A ps ) )
7		rmo5 2717	. 2 |- ( E* x e. A ph <-> ( E. x e. A ph -> E! x e. A ph ) )
8		rmo5 2717	. 2 |- ( E* y e. A ps <-> ( E. y e. A ps -> E! y e. A ps ) )
9	6, 7, 8	3bitr4i 212	1 |- ( E* x e. A ph <-> E* y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1474   E.wrex 2476   E!wreu 2477   E*wrmo 2478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-reu 2482  df-rmo 2483

This theorem is referenced by:  cbvrmov  2732  cbvdisj  4020