Theorem cbvrexfw 2729

Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexf 2731 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 1530 and ax-bndl 1532 in the proof. (Contributed by FL, 27-Apr-2008.) (Revised by GG, 10-Jan-2024.)

Hypotheses

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

Assertion

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

Distinct variable group:   x, y

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

Proof of Theorem cbvrexfw

Step	Hyp	Ref	Expression
1		cbvrexfw.2	. . . . 5 |- F/_ y A
2	1	nfcri 2342	. . . 4 |- F/ y x e. A
3		cbvrexfw.3	. . . 4 |- F/ y ph
4	2, 3	nfan 1588	. . 3 |- F/ y ( x e. A /\ ph )
5		cbvrexfw.1	. . . . 5 |- F/_ x A
6	5	nfcri 2342	. . . 4 |- F/ x y e. A
7		cbvrexfw.4	. . . 4 |- F/ x ps
8	6, 7	nfan 1588	. . 3 |- F/ x ( y e. A /\ ps )
9		eleq1w 2266	. . . 4 |- ( x = y -> ( x e. A <-> y e. A ) )
10		cbvrexfw.5	. . . 4 |- ( x = y -> ( ph <-> ps ) )
11	9, 10	anbi12d 473	. . 3 |- ( x = y -> ( ( x e. A /\ ph ) <-> ( y e. A /\ ps ) ) )
12	4, 8, 11	cbvexv1 1775	. 2 |- ( E. x ( x e. A /\ ph ) <-> E. y ( y e. A /\ ps ) )
13		df-rex 2490	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
14		df-rex 2490	. 2 |- ( E. y e. A ps <-> E. y ( y e. A /\ ps ) )
15	12, 13, 14	3bitr4i 212	1 |- ( E. x e. A ph <-> E. y e. A ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   F/wnf 1483   E.wex 1515   e. wcel 2176   F/_wnfc 2335   E.wrex 2485

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490

This theorem is referenced by:  cbvrexw  2733  nnwofdc  12392