Theorem cbvreuw 2760

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 2763 with a disjoint variable condition. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by GG, 10-Jan-2024.) (Revised by Wolf Lammen, 10-Dec-2024.)

Hypotheses

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

Assertion

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

Distinct variable group:   x, A, y

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

Proof of Theorem cbvreuw

Step	Hyp	Ref	Expression
1		cbvreuw.1	. . . 4 |- F/ y ph
2		cbvreuw.2	. . . 4 |- F/ x ps
3		cbvreuw.3	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvrexw 2759	. . 3 |- ( E. x e. A ph <-> E. y e. A ps )
5	1, 2, 3	cbvrmow 2714	. . 3 |- ( E* x e. A ph <-> E* y e. A ps )
6	4, 5	anbi12i 460	. 2 |- ( ( E. x e. A ph /\ E* x e. A ph ) <-> ( E. y e. A ps /\ E* y e. A ps ) )
7		reu5 2749	. 2 |- ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) )
8		reu5 2749	. 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   /\ wa 104   <-> wb 105   F/wnf 1506   E.wrex 2509   E!wreu 2510   E*wrmo 2511

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-reu 2515  df-rmo 2516

This theorem is referenced by:  reu8nf  3110