Theorem cbvrabcsf 3202

Description: A more general version of cbvrab 2858 with no distinct variable restrictions. (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
cbvrabcsf	|- {x e. A | ph} = {y e. B | ps}

Proof of Theorem cbvrabcsf

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

Step	Hyp	Ref	Expression
1		nfv 1619	. . . 4 |- F/z(x e. A /\ ph)
2		nfcsb1v 3169	. . . . . 6 |- F/_x[_z / x]_A
3	2	nfcri 2484	. . . . 5 |- F/x z e. [_z / x]_A
4		nfs1v 2106	. . . . 5 |- F/x[z / x]ph
5	3, 4	nfan 1824	. . . 4 |- F/x(z e. [_z / x]_A /\ [z / x]ph)
6		id 19	. . . . . 6 |- (x = z -> x = z)
7		csbeq1a 3145	. . . . . 6 |- (x = z -> A = [_z / x]_A)
8	6, 7	eleq12d 2421	. . . . 5 |- (x = z -> (x e. A <-> z e. [_z / x]_A))
9		sbequ12 1919	. . . . 5 |- (x = z -> (ph <-> [z / x]ph))
10	8, 9	anbi12d 691	. . . 4 |- (x = z -> ((x e. A /\ ph) <-> (z e. [_z / x]_A /\ [z / x]ph)))
11	1, 5, 10	cbvab 2472	. . 3 |- {x | (x e. A /\ ph)} = {z | (z e. [_z / x]_A /\ [z / x]ph)}
12		nfcv 2490	. . . . . . 7 |- F/_yz
13		cbvralcsf.1	. . . . . . 7 |- F/_yA
14	12, 13	nfcsb 3171	. . . . . 6 |- F/_y[_z / x]_A
15	14	nfcri 2484	. . . . 5 |- F/y z e. [_z / x]_A
16		cbvralcsf.3	. . . . . 6 |- F/yph
17	16	nfsb 2109	. . . . 5 |- F/y[z / x]ph
18	15, 17	nfan 1824	. . . 4 |- F/y(z e. [_z / x]_A /\ [z / x]ph)
19		nfv 1619	. . . 4 |- F/z(y e. B /\ ps)
20		id 19	. . . . . 6 |- (z = y -> z = y)
21		csbeq1 3140	. . . . . . 7 |- (z = y -> [_z / x]_A = [_y / x]_A)
22		df-csb 3138	. . . . . . . 8 |- [_y / x]_A = {v | [.y / x ].v e. A}
23		cbvralcsf.2	. . . . . . . . . . . 12 |- F/_xB
24	23	nfcri 2484	. . . . . . . . . . 11 |- F/x v e. B
25		cbvralcsf.5	. . . . . . . . . . . 12 |- (x = y -> A = B)
26	25	eleq2d 2420	. . . . . . . . . . 11 |- (x = y -> (v e. A <-> v e. B))
27	24, 26	sbie 2038	. . . . . . . . . 10 |- ([y / x]v e. A <-> v e. B)
28		sbsbc 3051	. . . . . . . . . 10 |- ([y / x]v e. A <-> [.y / x ].v e. A)
29	27, 28	bitr3i 242	. . . . . . . . 9 |- (v e. B <-> [.y / x ].v e. A)
30	29	eqabi 2465	. . . . . . . 8 |- B = {v | [.y / x ].v e. A}
31	22, 30	eqtr4i 2376	. . . . . . 7 |- [_y / x]_A = B
32	21, 31	syl6eq 2401	. . . . . 6 |- (z = y -> [_z / x]_A = B)
33	20, 32	eleq12d 2421	. . . . 5 |- (z = y -> (z e. [_z / x]_A <-> y e. B))
34		sbequ 2060	. . . . . 6 |- (z = y -> ([z / x]ph <-> [y / x]ph))
35		cbvralcsf.4	. . . . . . 7 |- F/xps
36		cbvralcsf.6	. . . . . . 7 |- (x = y -> (ph <-> ps))
37	35, 36	sbie 2038	. . . . . 6 |- ([y / x]ph <-> ps)
38	34, 37	syl6bb 252	. . . . 5 |- (z = y -> ([z / x]ph <-> ps))
39	33, 38	anbi12d 691	. . . 4 |- (z = y -> ((z e. [_z / x]_A /\ [z / x]ph) <-> (y e. B /\ ps)))
40	18, 19, 39	cbvab 2472	. . 3 |- {z | (z e. [_z / x]_A /\ [z / x]ph)} = {y | (y e. B /\ ps)}
41	11, 40	eqtri 2373	. 2 |- {x | (x e. A /\ ph)} = {y | (y e. B /\ ps)}
42		df-rab 2624	. 2 |- {x e. A | ph} = {x | (x e. A /\ ph)}
43		df-rab 2624	. 2 |- {y e. B | ps} = {y | (y e. B /\ ps)}
44	41, 42, 43	3eqtr4i 2383	1 |- {x e. A | ph} = {y e. B | ps}

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   F/wnf 1544   = wceq 1642  [wsb 1648   e. wcel 1710  {cab 2339   F/_wnfc 2477  {crab 2619   [.wsbc 3047  [_csb 3137

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-rab 2624  df-sbc 3048  df-csb 3138

This theorem is referenced by: (None)