Theorem cbvreuw 2781

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 2784 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 2780	. . 3 |- ( E. x e. A ph <-> E. y e. A ps )
5	1, 2, 3	cbvrmow 2735	. . 3 |- ( E* x e. A ph <-> E* y e. A ps )
6	4, 5	anbi12i 464	. 2 |- ( ( E. x e. A ph /\ E* x e. A ph ) <-> ( E. y e. A ps /\ E* y e. A ps ) )
7		reu5 2770	. 2 |- ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) )
8		reu5 2770	. 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 1513   E.wrex 2529   E!wreu 2530   E*wrmo 2531

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-reu 2535  df-rmo 2536

This theorem is referenced by:  reu8nf  3133