Theorem cbvrmo 3170

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 3168	. . 3 |- ( E. x e. A ph <-> E. y e. A ps )
5	1, 2, 3	cbvreu 3169	. . 3 |- ( E! x e. A ph <-> E! y e. A ps )
6	4, 5	imbi12i 340	. 2 |- ( ( E. x e. A ph -> E! x e. A ph ) <-> ( E. y e. A ps -> E! y e. A ps ) )
7		rmo5 3162	. 2 |- ( E* x e. A ph <-> ( E. x e. A ph -> E! x e. A ph ) )
8		rmo5 3162	. 2 |- ( E* y e. A ps <-> ( E. y e. A ps -> E! y e. A ps ) )
9	6, 7, 8	3bitr4i 292	1 |- ( E* x e. A ph <-> E* y e. A ps )

Syntax hints:   -> wi 4   <-> wb 196   F/wnf 1708   E.wrex 2913   E!wreu 2914   E*wrmo 2915

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-eu 2474  df-mo 2475  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920

This theorem is referenced by:  cbvrmov  3174  cbvdisj  4630