Theorem cbvrexcsf 3200

Description: A more general version of cbvrexf 2831 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)

Hypotheses

Ref	Expression
cbvralcsf.1	|- F/_yA
cbvralcsf.2	|- F/_xB
cbvralcsf.3	|- F/yph
cbvralcsf.4	|- F/xps
cbvralcsf.5	|- (x = y -> A = B)
cbvralcsf.6	|- (x = y -> (ph <-> ps))

Assertion

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

Proof of Theorem cbvrexcsf

Step	Hyp	Ref	Expression
1		cbvralcsf.1	. . . 4 |- F/_yA
2		cbvralcsf.2	. . . 4 |- F/_xB
3		cbvralcsf.3	. . . . 5 |- F/yph
4	3	nfn 1793	. . . 4 |- F/y -. ph
5		cbvralcsf.4	. . . . 5 |- F/xps
6	5	nfn 1793	. . . 4 |- F/x -. ps
7		cbvralcsf.5	. . . 4 |- (x = y -> A = B)
8		cbvralcsf.6	. . . . 5 |- (x = y -> (ph <-> ps))
9	8	notbid 285	. . . 4 |- (x = y -> (-. ph <-> -. ps))
10	1, 2, 4, 6, 7, 9	cbvralcsf 3199	. . 3 |- (A.x e. A -. ph <-> A.y e. B -. ps)
11	10	notbii 287	. 2 |- (-. A.x e. A -. ph <-> -. A.y e. B -. ps)
12		dfrex2 2628	. 2 |- (E.x e. A ph <-> -. A.x e. A -. ph)
13		dfrex2 2628	. 2 |- (E.y e. B ps <-> -. A.y e. B -. ps)
14	11, 12, 13	3bitr4i 268	1 |- (E.x e. A ph <-> E.y e. B ps)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   F/wnf 1544   = wceq 1642   F/_wnfc 2477  A.wral 2615  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-sbc 3048  df-csb 3138

This theorem is referenced by:  cbvrexv2  3204