Theorem cbvreuvw 2735

Description: Version of cbvreuv 2731 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 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 2257	. . . . . . 7 |- ( x = y -> ( x e. A <-> y e. A ) )
2		cbvralvw.1	. . . . . . 7 |- ( x = y -> ( ph <-> ps ) )
3	1, 2	anbi12d 473	. . . . . 6 |- ( x = y -> ( ( x e. A /\ ph ) <-> ( y e. A /\ ps ) ) )
4		equequ1 1726	. . . . . 6 |- ( x = y -> ( x = z <-> y = z ) )
5	3, 4	bibi12d 235	. . . . 5 |- ( x = y -> ( ( ( x e. A /\ ph ) <-> x = z ) <-> ( ( y e. A /\ ps ) <-> y = z ) ) )
6	5	cbvalvw 1934	. . . 4 |- ( A. x ( ( x e. A /\ ph ) <-> x = z ) <-> A. y ( ( y e. A /\ ps ) <-> y = z ) )
7	6	exbii 1619	. . 3 |- ( E. z A. x ( ( x e. A /\ ph ) <-> x = z ) <-> E. z A. y ( ( y e. A /\ ps ) <-> y = z ) )
8		df-eu 2048	. . 3 |- ( E! x ( x e. A /\ ph ) <-> E. z A. x ( ( x e. A /\ ph ) <-> x = z ) )
9		df-eu 2048	. . 3 |- ( E! y ( y e. A /\ ps ) <-> E. z A. y ( ( y e. A /\ ps ) <-> y = z ) )
10	7, 8, 9	3bitr4ri 213	. 2 |- ( E! y ( y e. A /\ ps ) <-> E! x ( x e. A /\ ph ) )
11		df-reu 2482	. 2 |- ( E! y e. A ps <-> E! y ( y e. A /\ ps ) )
12		df-reu 2482	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
13	10, 11, 12	3bitr4ri 213	1 |- ( E! x e. A ph <-> E! y e. A ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   E.wex 1506   E!weu 2045   e. wcel 2167   E!wreu 2477

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-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-eu 2048  df-clel 2192  df-reu 2482

This theorem is referenced by: (None)