Theorem cbvrabcsf 3031

Description: A more general version of cbvrab 2655 with no distinct variable restrictions. (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
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 1491	. . . 4 |- F/ z ( x e. A /\ ph )
2		nfcsb1v 3001	. . . . . 6 |- F/_ x [_ z / x ]_ A
3	2	nfcri 2249	. . . . 5 |- F/ x z e. [_ z / x ]_ A
4		nfs1v 1890	. . . . 5 |- F/ x [ z / x ] ph
5	3, 4	nfan 1527	. . . 4 |- F/ x ( z e. [_ z / x ]_ A /\ [ z / x ] ph )
6		id 19	. . . . . 6 |- ( x = z -> x = z )
7		csbeq1a 2979	. . . . . 6 |- ( x = z -> A = [_ z / x ]_ A )
8	6, 7	eleq12d 2185	. . . . 5 |- ( x = z -> ( x e. A <-> z e. [_ z / x ]_ A ) )
9		sbequ12 1727	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
10	8, 9	anbi12d 462	. . . 4 |- ( x = z -> ( ( x e. A /\ ph ) <-> ( z e. [_ z / x ]_ A /\ [ z / x ] ph ) ) )
11	1, 5, 10	cbvab 2237	. . 3 |- { x | ( x e. A /\ ph ) } = { z | ( z e. [_ z / x ]_ A /\ [ z / x ] ph ) }
12		nfcv 2255	. . . . . . 7 |- F/_ y z
13		cbvralcsf.1	. . . . . . 7 |- F/_ y A
14	12, 13	nfcsb 3003	. . . . . 6 |- F/_ y [_ z / x ]_ A
15	14	nfcri 2249	. . . . 5 |- F/ y z e. [_ z / x ]_ A
16		cbvralcsf.3	. . . . . 6 |- F/ y ph
17	16	nfsb 1897	. . . . 5 |- F/ y [ z / x ] ph
18	15, 17	nfan 1527	. . . 4 |- F/ y ( z e. [_ z / x ]_ A /\ [ z / x ] ph )
19		nfv 1491	. . . 4 |- F/ z ( y e. B /\ ps )
20		id 19	. . . . . 6 |- ( z = y -> z = y )
21		csbeq1 2974	. . . . . . 7 |- ( z = y -> [_ z / x ]_ A = [_ y / x ]_ A )
22		df-csb 2972	. . . . . . . 8 |- [_ y / x ]_ A = { v | [. y / x ]. v e. A }
23		cbvralcsf.2	. . . . . . . . . . . 12 |- F/_ x B
24	23	nfcri 2249	. . . . . . . . . . 11 |- F/ x v e. B
25		cbvralcsf.5	. . . . . . . . . . . 12 |- ( x = y -> A = B )
26	25	eleq2d 2184	. . . . . . . . . . 11 |- ( x = y -> ( v e. A <-> v e. B ) )
27	24, 26	sbie 1747	. . . . . . . . . 10 |- ( [ y / x ] v e. A <-> v e. B )
28		sbsbc 2882	. . . . . . . . . 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 2229	. . . . . . . 8 |- B = { v | [. y / x ]. v e. A }
31	22, 30	eqtr4i 2138	. . . . . . 7 |- [_ y / x ]_ A = B
32	21, 31	syl6eq 2163	. . . . . 6 |- ( z = y -> [_ z / x ]_ A = B )
33	20, 32	eleq12d 2185	. . . . 5 |- ( z = y -> ( z e. [_ z / x ]_ A <-> y e. B ) )
34		sbequ 1794	. . . . . 6 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
35		cbvralcsf.4	. . . . . . 7 |- F/ x ps
36		cbvralcsf.6	. . . . . . 7 |- ( x = y -> ( ph <-> ps ) )
37	35, 36	sbie 1747	. . . . . 6 |- ( [ y / x ] ph <-> ps )
38	34, 37	syl6bb 195	. . . . 5 |- ( z = y -> ( [ z / x ] ph <-> ps ) )
39	33, 38	anbi12d 462	. . . 4 |- ( z = y -> ( ( z e. [_ z / x ]_ A /\ [ z / x ] ph ) <-> ( y e. B /\ ps ) ) )
40	18, 19, 39	cbvab 2237	. . 3 |- { z | ( z e. [_ z / x ]_ A /\ [ z / x ] ph ) } = { y | ( y e. B /\ ps ) }
41	11, 40	eqtri 2135	. 2 |- { x | ( x e. A /\ ph ) } = { y | ( y e. B /\ ps ) }
42		df-rab 2399	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
43		df-rab 2399	. 2 |- { y e. B | ps } = { y | ( y e. B /\ ps ) }
44	41, 42, 43	3eqtr4i 2145	1 |- { x e. A | ph } = { y e. B | ps }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1314   F/wnf 1419   e. wcel 1463   [wsb 1718   {cab 2101   F/_wnfc 2242   {crab 2394   [.wsbc 2878   [_csb 2971

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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rab 2399  df-sbc 2879  df-csb 2972

This theorem is referenced by: (None)