Theorem cbvreuw 2762

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 2765 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 2761	. . 3 |- ( E. x e. A ph <-> E. y e. A ps )
5	1, 2, 3	cbvrmow 2716	. . 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 2751	. 2 |- ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) )
8		reu5 2751	. 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 1508   E.wrex 2511   E!wreu 2512   E*wrmo 2513

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-reu 2517  df-rmo 2518

This theorem is referenced by:  reu8nf  3113