Theorem cbvreuvw 2698

Description: Version of cbvreuv 2694 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)

Hypothesis

Ref	Expression
cbvralvw.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   x, y, A   ph, y   ps, x

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

Proof of Theorem cbvreuvw

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2227	. . . . . . 7 |- ( x = y -> ( x e. A <-> y e. A ) )
2		cbvralvw.1	. . . . . . 7 |- ( x = y -> ( ph <-> ps ) )
3	1, 2	anbi12d 465	. . . . . 6 |- ( x = y -> ( ( x e. A /\ ph ) <-> ( y e. A /\ ps ) ) )
4		equequ1 1700	. . . . . 6 |- ( x = y -> ( x = z <-> y = z ) )
5	3, 4	bibi12d 234	. . . . 5 |- ( x = y -> ( ( ( x e. A /\ ph ) <-> x = z ) <-> ( ( y e. A /\ ps ) <-> y = z ) ) )
6	5	cbvalvw 1907	. . . 4 |- ( A. x ( ( x e. A /\ ph ) <-> x = z ) <-> A. y ( ( y e. A /\ ps ) <-> y = z ) )
7	6	exbii 1593	. . 3 |- ( E. z A. x ( ( x e. A /\ ph ) <-> x = z ) <-> E. z A. y ( ( y e. A /\ ps ) <-> y = z ) )
8		df-eu 2017	. . 3 |- ( E! x ( x e. A /\ ph ) <-> E. z A. x ( ( x e. A /\ ph ) <-> x = z ) )
9		df-eu 2017	. . 3 |- ( E! y ( y e. A /\ ps ) <-> E. z A. y ( ( y e. A /\ ps ) <-> y = z ) )
10	7, 8, 9	3bitr4ri 212	. 2 |- ( E! y ( y e. A /\ ps ) <-> E! x ( x e. A /\ ph ) )
11		df-reu 2451	. 2 |- ( E! y e. A ps <-> E! y ( y e. A /\ ps ) )
12		df-reu 2451	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
13	10, 11, 12	3bitr4ri 212	1 |- ( E! x e. A ph <-> E! y e. A ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   E.wex 1480   E!weu 2014   e. wcel 2136   E!wreu 2446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-eu 2017  df-clel 2161  df-reu 2451

This theorem is referenced by: (None)