Theorem cbvrexcsf 3013

Description: A more general version of cbvrexf 2607 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/_ y A
cbvralcsf.2	|- F/_ x B
cbvralcsf.3	|- F/ y ph
cbvralcsf.4	|- F/ x ps
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

Dummy variables v z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1476	. . . 4 |- F/ z ( x e. A /\ ph )
2		nfcsb1v 2985	. . . . . 6 |- F/_ x [_ z / x ]_ A
3	2	nfcri 2234	. . . . 5 |- F/ x z e. [_ z / x ]_ A
4		nfsbc1v 2880	. . . . 5 |- F/ x [. z / x ]. ph
5	3, 4	nfan 1512	. . . 4 |- F/ x ( z e. [_ z / x ]_ A /\ [. z / x ]. ph )
6		id 19	. . . . . 6 |- ( x = z -> x = z )
7		csbeq1a 2963	. . . . . 6 |- ( x = z -> A = [_ z / x ]_ A )
8	6, 7	eleq12d 2170	. . . . 5 |- ( x = z -> ( x e. A <-> z e. [_ z / x ]_ A ) )
9		sbceq1a 2871	. . . . 5 |- ( x = z -> ( ph <-> [. z / x ]. ph ) )
10	8, 9	anbi12d 460	. . . 4 |- ( x = z -> ( ( x e. A /\ ph ) <-> ( z e. [_ z / x ]_ A /\ [. z / x ]. ph ) ) )
11	1, 5, 10	cbvex 1697	. . 3 |- ( E. x ( x e. A /\ ph ) <-> E. z ( z e. [_ z / x ]_ A /\ [. z / x ]. ph ) )
12		nfcv 2240	. . . . . . 7 |- F/_ y z
13		cbvralcsf.1	. . . . . . 7 |- F/_ y A
14	12, 13	nfcsb 2987	. . . . . 6 |- F/_ y [_ z / x ]_ A
15	14	nfcri 2234	. . . . 5 |- F/ y z e. [_ z / x ]_ A
16		cbvralcsf.3	. . . . . 6 |- F/ y ph
17	12, 16	nfsbc 2882	. . . . 5 |- F/ y [. z / x ]. ph
18	15, 17	nfan 1512	. . . 4 |- F/ y ( z e. [_ z / x ]_ A /\ [. z / x ]. ph )
19		nfv 1476	. . . 4 |- F/ z ( y e. B /\ ps )
20		id 19	. . . . . 6 |- ( z = y -> z = y )
21		csbeq1 2958	. . . . . . 7 |- ( z = y -> [_ z / x ]_ A = [_ y / x ]_ A )
22		df-csb 2956	. . . . . . . 8 |- [_ y / x ]_ A = { v | [. y / x ]. v e. A }
23		cbvralcsf.2	. . . . . . . . . . . 12 |- F/_ x B
24	23	nfcri 2234	. . . . . . . . . . 11 |- F/ x v e. B
25		cbvralcsf.5	. . . . . . . . . . . 12 |- ( x = y -> A = B )
26	25	eleq2d 2169	. . . . . . . . . . 11 |- ( x = y -> ( v e. A <-> v e. B ) )
27	24, 26	sbie 1732	. . . . . . . . . 10 |- ( [ y / x ] v e. A <-> v e. B )
28		sbsbc 2866	. . . . . . . . . 10 |- ( [ y / x ] v e. A <-> [. y / x ]. v e. A )
29	27, 28	bitr3i 185	. . . . . . . . 9 |- ( v e. B <-> [. y / x ]. v e. A )
30	29	abbi2i 2214	. . . . . . . 8 |- B = { v | [. y / x ]. v e. A }
31	22, 30	eqtr4i 2123	. . . . . . 7 |- [_ y / x ]_ A = B
32	21, 31	syl6eq 2148	. . . . . 6 |- ( z = y -> [_ z / x ]_ A = B )
33	20, 32	eleq12d 2170	. . . . 5 |- ( z = y -> ( z e. [_ z / x ]_ A <-> y e. B ) )
34		dfsbcq 2864	. . . . . 6 |- ( z = y -> ( [. z / x ]. ph <-> [. y / x ]. ph ) )
35		sbsbc 2866	. . . . . . 7 |- ( [ y / x ] ph <-> [. y / x ]. ph )
36		cbvralcsf.4	. . . . . . . 8 |- F/ x ps
37		cbvralcsf.6	. . . . . . . 8 |- ( x = y -> ( ph <-> ps ) )
38	36, 37	sbie 1732	. . . . . . 7 |- ( [ y / x ] ph <-> ps )
39	35, 38	bitr3i 185	. . . . . 6 |- ( [. y / x ]. ph <-> ps )
40	34, 39	syl6bb 195	. . . . 5 |- ( z = y -> ( [. z / x ]. ph <-> ps ) )
41	33, 40	anbi12d 460	. . . 4 |- ( z = y -> ( ( z e. [_ z / x ]_ A /\ [. z / x ]. ph ) <-> ( y e. B /\ ps ) ) )
42	18, 19, 41	cbvex 1697	. . 3 |- ( E. z ( z e. [_ z / x ]_ A /\ [. z / x ]. ph ) <-> E. y ( y e. B /\ ps ) )
43	11, 42	bitri 183	. 2 |- ( E. x ( x e. A /\ ph ) <-> E. y ( y e. B /\ ps ) )
44		df-rex 2381	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
45		df-rex 2381	. 2 |- ( E. y e. B ps <-> E. y ( y e. B /\ ps ) )
46	43, 44, 45	3bitr4i 211	1 |- ( E. x e. A ph <-> E. y e. B ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1299   F/wnf 1404   E.wex 1436   e. wcel 1448   [wsb 1703   {cab 2086   F/_wnfc 2227   E.wrex 2376   [.wsbc 2862   [_csb 2955

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-sbc 2863  df-csb 2956

This theorem is referenced by:  cbvrexv2  3017