Theorem cbvralcsf 3210

Description: A more general version of cbvralf 2777 that doesn't require A and B to be distinct from x or y. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)

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
cbvralcsf	|- ( A. x e. A ph <-> A. y e. B ps )

Proof of Theorem cbvralcsf

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

Step	Hyp	Ref	Expression
1		nfv 1581	. . . 4 |- F/ z ( x e. A -> ph )
2		nfcsb1v 3180	. . . . . 6 |- F/_ x [_ z / x ]_ A
3	2	nfcri 2386	. . . . 5 |- F/ x z e. [_ z / x ]_ A
4		nfsbc1v 3070	. . . . 5 |- F/ x [. z / x ]. ph
5	3, 4	nfim 1625	. . . 4 |- F/ x ( z e. [_ z / x ]_ A -> [. z / x ]. ph )
6		id 19	. . . . . 6 |- ( x = z -> x = z )
7		csbeq1a 3156	. . . . . 6 |- ( x = z -> A = [_ z / x ]_ A )
8	6, 7	eleq12d 2309	. . . . 5 |- ( x = z -> ( x e. A <-> z e. [_ z / x ]_ A ) )
9		sbceq1a 3061	. . . . 5 |- ( x = z -> ( ph <-> [. z / x ]. ph ) )
10	8, 9	imbi12d 234	. . . 4 |- ( x = z -> ( ( x e. A -> ph ) <-> ( z e. [_ z / x ]_ A -> [. z / x ]. ph ) ) )
11	1, 5, 10	cbval 1807	. . 3 |- ( A. x ( x e. A -> ph ) <-> A. z ( z e. [_ z / x ]_ A -> [. z / x ]. ph ) )
12		nfcv 2392	. . . . . . 7 |- F/_ y z
13		cbvralcsf.1	. . . . . . 7 |- F/_ y A
14	12, 13	nfcsb 3185	. . . . . 6 |- F/_ y [_ z / x ]_ A
15	14	nfcri 2386	. . . . 5 |- F/ y z e. [_ z / x ]_ A
16		cbvralcsf.3	. . . . . 6 |- F/ y ph
17	12, 16	nfsbc 3072	. . . . 5 |- F/ y [. z / x ]. ph
18	15, 17	nfim 1625	. . . 4 |- F/ y ( z e. [_ z / x ]_ A -> [. z / x ]. ph )
19		nfv 1581	. . . 4 |- F/ z ( y e. B -> ps )
20		id 19	. . . . . 6 |- ( z = y -> z = y )
21		csbeq1 3150	. . . . . . 7 |- ( z = y -> [_ z / x ]_ A = [_ y / x ]_ A )
22		df-csb 3148	. . . . . . . 8 |- [_ y / x ]_ A = { v | [. y / x ]. v e. A }
23		cbvralcsf.2	. . . . . . . . . . . 12 |- F/_ x B
24	23	nfcri 2386	. . . . . . . . . . 11 |- F/ x v e. B
25		cbvralcsf.5	. . . . . . . . . . . 12 |- ( x = y -> A = B )
26	25	eleq2d 2308	. . . . . . . . . . 11 |- ( x = y -> ( v e. A <-> v e. B ) )
27	24, 26	sbie 1844	. . . . . . . . . 10 |- ( [ y / x ] v e. A <-> v e. B )
28		sbsbc 3055	. . . . . . . . . 10 |- ( [ y / x ] v e. A <-> [. y / x ]. v e. A )
29	27, 28	bitr3i 186	. . . . . . . . 9 |- ( v e. B <-> [. y / x ]. v e. A )
30	29	abbi2i 2353	. . . . . . . 8 |- B = { v | [. y / x ]. v e. A }
31	22, 30	eqtr4i 2262	. . . . . . 7 |- [_ y / x ]_ A = B
32	21, 31	eqtrdi 2287	. . . . . 6 |- ( z = y -> [_ z / x ]_ A = B )
33	20, 32	eleq12d 2309	. . . . 5 |- ( z = y -> ( z e. [_ z / x ]_ A <-> y e. B ) )
34		dfsbcq 3053	. . . . . 6 |- ( z = y -> ( [. z / x ]. ph <-> [. y / x ]. ph ) )
35		sbsbc 3055	. . . . . . 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 1844	. . . . . . 7 |- ( [ y / x ] ph <-> ps )
39	35, 38	bitr3i 186	. . . . . 6 |- ( [. y / x ]. ph <-> ps )
40	34, 39	bitrdi 196	. . . . 5 |- ( z = y -> ( [. z / x ]. ph <-> ps ) )
41	33, 40	imbi12d 234	. . . 4 |- ( z = y -> ( ( z e. [_ z / x ]_ A -> [. z / x ]. ph ) <-> ( y e. B -> ps ) ) )
42	18, 19, 41	cbval 1807	. . 3 |- ( A. z ( z e. [_ z / x ]_ A -> [. z / x ]. ph ) <-> A. y ( y e. B -> ps ) )
43	11, 42	bitri 184	. 2 |- ( A. x ( x e. A -> ph ) <-> A. y ( y e. B -> ps ) )
44		df-ral 2533	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
45		df-ral 2533	. 2 |- ( A. y e. B ps <-> A. y ( y e. B -> ps ) )
46	43, 44, 45	3bitr4i 212	1 |- ( A. x e. A ph <-> A. y e. B ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1400   = wceq 1402   F/wnf 1513   [wsb 1815   e. wcel 2209   {cab 2224   F/_wnfc 2379   A.wral 2528   [.wsbc 3051   [_csb 3147

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-sbc 3052  df-csb 3148

This theorem is referenced by:  cbvralv2  3214