Theorem cbvralcsf 2902

Description: A more general version of cbvralf 2521 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/_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
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 1418	. . . 4 |- F/z(x e. A -> ph)
2		nfcsb1v 2876	. . . . . 6 |- F/_x [_z / x ]_A
3	2	nfcri 2169	. . . . 5 |- F/x z e. [_z / x ]_A
4		nfsbc1v 2776	. . . . 5 |- F/x [.z / x ].ph
5	3, 4	nfim 1461	. . . 4 |- F/x(z e. [_z / x ]_A -> [.z / x ].ph)
6		id 19	. . . . . 6 |- (x = z -> x = z)
7		csbeq1a 2854	. . . . . 6 |- (x = z -> A = [_z / x ]_A)
8	6, 7	eleq12d 2105	. . . . 5 |- (x = z -> (x e. A <-> z e. [_z / x ]_A))
9		sbceq1a 2767	. . . . 5 |- (x = z -> (ph <-> [.z / x ].ph))
10	8, 9	imbi12d 223	. . . 4 |- (x = z -> ((x e. A -> ph) <-> (z e. [_z / x ]_A -> [.z / x ].ph)))
11	1, 5, 10	cbval 1634	. . 3 |- (A.x(x e. A -> ph) <-> A.z(z e. [_z / x ]_A -> [.z / x ].ph))
12		nfcv 2175	. . . . . . 7 |- F/_yz
13		cbvralcsf.1	. . . . . . 7 |- F/_yA
14	12, 13	nfcsb 2878	. . . . . 6 |- F/_y [_z / x ]_A
15	14	nfcri 2169	. . . . 5 |- F/y z e. [_z / x ]_A
16		cbvralcsf.3	. . . . . 6 |- F/yph
17	12, 16	nfsbc 2778	. . . . 5 |- F/y [.z / x ].ph
18	15, 17	nfim 1461	. . . 4 |- F/y(z e. [_z / x ]_A -> [.z / x ].ph)
19		nfv 1418	. . . 4 |- F/z(y e. B -> ps)
20		id 19	. . . . . 6 |- (z = y -> z = y)
21		csbeq1 2849	. . . . . . 7 |- (z = y -> [_z / x ]_A = [_y / x ]_A)
22		df-csb 2847	. . . . . . . 8 |- [_y / x ]_A = {v | [.y / x ].v e. A }
23		cbvralcsf.2	. . . . . . . . . . . 12 |- F/_xB
24	23	nfcri 2169	. . . . . . . . . . 11 |- F/x v e. B
25		cbvralcsf.5	. . . . . . . . . . . 12 |- (x = y -> A = B)
26	25	eleq2d 2104	. . . . . . . . . . 11 |- (x = y -> (v e. A <-> v e. B))
27	24, 26	sbie 1671	. . . . . . . . . 10 |- ([y / x]v e. A <-> v e. B)
28		sbsbc 2762	. . . . . . . . . 10 |- ([y / x]v e. A <-> [.y / x ].v e. A)
29	27, 28	bitr3i 175	. . . . . . . . 9 |- (v e. B <-> [.y / x ].v e. A)
30	29	abbi2i 2149	. . . . . . . 8 |- B = {v | [.y / x ].v e. A }
31	22, 30	eqtr4i 2060	. . . . . . 7 |- [_y / x ]_A = B
32	21, 31	syl6eq 2085	. . . . . 6 |- (z = y -> [_z / x ]_A = B)
33	20, 32	eleq12d 2105	. . . . 5 |- (z = y -> (z e. [_z / x ]_A <-> y e. B))
34		dfsbcq 2760	. . . . . 6 |- (z = y -> ( [.z / x ].ph <-> [.y / x ].ph))
35		sbsbc 2762	. . . . . . 7 |- ([y / x]ph <-> [.y / x ].ph)
36		cbvralcsf.4	. . . . . . . 8 |- F/xps
37		cbvralcsf.6	. . . . . . . 8 |- (x = y -> (ph <-> ps))
38	36, 37	sbie 1671	. . . . . . 7 |- ([y / x]ph <-> ps)
39	35, 38	bitr3i 175	. . . . . 6 |- ( [.y / x ].ph <-> ps)
40	34, 39	syl6bb 185	. . . . 5 |- (z = y -> ( [.z / x ].ph <-> ps))
41	33, 40	imbi12d 223	. . . 4 |- (z = y -> ((z e. [_z / x ]_A -> [.z / x ].ph) <-> (y e. B -> ps)))
42	18, 19, 41	cbval 1634	. . 3 |- (A.z(z e. [_z / x ]_A -> [.z / x ].ph) <-> A.y(y e. B -> ps))
43	11, 42	bitri 173	. 2 |- (A.x(x e. A -> ph) <-> A.y(y e. B -> ps))
44		df-ral 2305	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
45		df-ral 2305	. 2 |- (A.y e. B ps <-> A.y(y e. B -> ps))
46	43, 44, 45	3bitr4i 201	1 |- (A.x e. A ph <-> A.y e. B ps)

Syntax hints:   -> wi 4   <-> wb 98  A.wal 1240   = wceq 1242   F/wnf 1346   e. wcel 1390  [wsb 1642   {cab 2023   F/_wnfc 2162  A.wral 2300   [.wsbc 2758   [_csb 2846

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-sbc 2759  df-csb 2847

This theorem is referenced by:  cbvralv2  2906