Theorem cbvralfw 2754

Description: Rule used to change bound variables, using implicit substitution. Version of cbvralf 2756 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1553 and ax-bndl 1555 in the proof. (Contributed by NM, 7-Mar-2004.) (Revised by GG, 23-May-2024.)

Hypotheses

Ref	Expression
cbvralfw.1	|- F/_ x A
cbvralfw.2	|- F/_ y A
cbvralfw.3	|- F/ y ph
cbvralfw.4	|- F/ x ps
cbvralfw.5	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvralfw	|- ( A. x e. A ph <-> A. y e. A ps )

Distinct variable group:   x, y

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

Proof of Theorem cbvralfw

Step	Hyp	Ref	Expression
1		cbvralfw.2	. . . . 5 |- F/_ y A
2	1	nfcri 2366	. . . 4 |- F/ y x e. A
3		cbvralfw.3	. . . 4 |- F/ y ph
4	2, 3	nfim 1618	. . 3 |- F/ y ( x e. A -> ph )
5		cbvralfw.1	. . . . 5 |- F/_ x A
6	5	nfcri 2366	. . . 4 |- F/ x y e. A
7		cbvralfw.4	. . . 4 |- F/ x ps
8	6, 7	nfim 1618	. . 3 |- F/ x ( y e. A -> ps )
9		eleq1w 2290	. . . 4 |- ( x = y -> ( x e. A <-> y e. A ) )
10		cbvralfw.5	. . . 4 |- ( x = y -> ( ph <-> ps ) )
11	9, 10	imbi12d 234	. . 3 |- ( x = y -> ( ( x e. A -> ph ) <-> ( y e. A -> ps ) ) )
12	4, 8, 11	cbvalv1 1797	. 2 |- ( A. x ( x e. A -> ph ) <-> A. y ( y e. A -> ps ) )
13		df-ral 2513	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
14		df-ral 2513	. 2 |- ( A. y e. A ps <-> A. y ( y e. A -> ps ) )
15	12, 13, 14	3bitr4i 212	1 |- ( A. x e. A ph <-> A. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   F/wnf 1506   e. wcel 2200   F/_wnfc 2359   A.wral 2508

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-nf 1507  df-sb 1809  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513

This theorem is referenced by:  cbvralw  2758  nnwofdc  12559