Theorem cbvralfw 2695

Description: Rule used to change bound variables, using implicit substitution. Version of cbvralf 2697 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 1507 and ax-bndl 1509 in the proof. (Contributed by NM, 7-Mar-2004.) (Revised by Gino Giotto, 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 2313	. . . 4 |- F/ y x e. A
3		cbvralfw.3	. . . 4 |- F/ y ph
4	2, 3	nfim 1572	. . 3 |- F/ y ( x e. A -> ph )
5		cbvralfw.1	. . . . 5 |- F/_ x A
6	5	nfcri 2313	. . . 4 |- F/ x y e. A
7		cbvralfw.4	. . . 4 |- F/ x ps
8	6, 7	nfim 1572	. . 3 |- F/ x ( y e. A -> ps )
9		eleq1w 2238	. . . 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 1751	. 2 |- ( A. x ( x e. A -> ph ) <-> A. y ( y e. A -> ps ) )
13		df-ral 2460	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
14		df-ral 2460	. 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 1351   F/wnf 1460   e. wcel 2148   F/_wnfc 2306   A.wral 2455

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460

This theorem is referenced by:  cbvralw  2699  nnwofdc  12042