Theorem cbvrabcsf 3568

Description: A more general version of cbvrab 3198 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 1843	. . . 4 |- F/ z ( x e. A /\ ph )
2		nfcsb1v 3549	. . . . . 6 |- F/_ x [_ z / x ]_ A
3	2	nfcri 2758	. . . . 5 |- F/ x z e. [_ z / x ]_ A
4		nfs1v 2437	. . . . 5 |- F/ x [ z / x ] ph
5	3, 4	nfan 1828	. . . 4 |- F/ x ( z e. [_ z / x ]_ A /\ [ z / x ] ph )
6		id 22	. . . . . 6 |- ( x = z -> x = z )
7		csbeq1a 3542	. . . . . 6 |- ( x = z -> A = [_ z / x ]_ A )
8	6, 7	eleq12d 2695	. . . . 5 |- ( x = z -> ( x e. A <-> z e. [_ z / x ]_ A ) )
9		sbequ12 2111	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
10	8, 9	anbi12d 747	. . . 4 |- ( x = z -> ( ( x e. A /\ ph ) <-> ( z e. [_ z / x ]_ A /\ [ z / x ] ph ) ) )
11	1, 5, 10	cbvab 2746	. . 3 |- { x | ( x e. A /\ ph ) } = { z | ( z e. [_ z / x ]_ A /\ [ z / x ] ph ) }
12		nfcv 2764	. . . . . . 7 |- F/_ y z
13		cbvralcsf.1	. . . . . . 7 |- F/_ y A
14	12, 13	nfcsb 3551	. . . . . 6 |- F/_ y [_ z / x ]_ A
15	14	nfcri 2758	. . . . 5 |- F/ y z e. [_ z / x ]_ A
16		cbvralcsf.3	. . . . . 6 |- F/ y ph
17	16	nfsb 2440	. . . . 5 |- F/ y [ z / x ] ph
18	15, 17	nfan 1828	. . . 4 |- F/ y ( z e. [_ z / x ]_ A /\ [ z / x ] ph )
19		nfv 1843	. . . 4 |- F/ z ( y e. B /\ ps )
20		id 22	. . . . . 6 |- ( z = y -> z = y )
21		csbeq1 3536	. . . . . . 7 |- ( z = y -> [_ z / x ]_ A = [_ y / x ]_ A )
22		df-csb 3534	. . . . . . . 8 |- [_ y / x ]_ A = { v | [. y / x ]. v e. A }
23		cbvralcsf.2	. . . . . . . . . . . 12 |- F/_ x B
24	23	nfcri 2758	. . . . . . . . . . 11 |- F/ x v e. B
25		cbvralcsf.5	. . . . . . . . . . . 12 |- ( x = y -> A = B )
26	25	eleq2d 2687	. . . . . . . . . . 11 |- ( x = y -> ( v e. A <-> v e. B ) )
27	24, 26	sbie 2408	. . . . . . . . . 10 |- ( [ y / x ] v e. A <-> v e. B )
28		sbsbc 3439	. . . . . . . . . 10 |- ( [ y / x ] v e. A <-> [. y / x ]. v e. A )
29	27, 28	bitr3i 266	. . . . . . . . 9 |- ( v e. B <-> [. y / x ]. v e. A )
30	29	abbi2i 2738	. . . . . . . 8 |- B = { v | [. y / x ]. v e. A }
31	22, 30	eqtr4i 2647	. . . . . . 7 |- [_ y / x ]_ A = B
32	21, 31	syl6eq 2672	. . . . . 6 |- ( z = y -> [_ z / x ]_ A = B )
33	20, 32	eleq12d 2695	. . . . 5 |- ( z = y -> ( z e. [_ z / x ]_ A <-> y e. B ) )
34		sbequ 2376	. . . . . 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 2408	. . . . . 6 |- ( [ y / x ] ph <-> ps )
38	34, 37	syl6bb 276	. . . . 5 |- ( z = y -> ( [ z / x ] ph <-> ps ) )
39	33, 38	anbi12d 747	. . . 4 |- ( z = y -> ( ( z e. [_ z / x ]_ A /\ [ z / x ] ph ) <-> ( y e. B /\ ps ) ) )
40	18, 19, 39	cbvab 2746	. . 3 |- { z | ( z e. [_ z / x ]_ A /\ [ z / x ] ph ) } = { y | ( y e. B /\ ps ) }
41	11, 40	eqtri 2644	. 2 |- { x | ( x e. A /\ ph ) } = { y | ( y e. B /\ ps ) }
42		df-rab 2921	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
43		df-rab 2921	. 2 |- { y e. B | ps } = { y | ( y e. B /\ ps ) }
44	41, 42, 43	3eqtr4i 2654	1 |- { x e. A | ph } = { y e. B | ps }

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   [wsb 1880   e. wcel 1990   {cab 2608   F/_wnfc 2751   {crab 2916   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-sbc 3436  df-csb 3534

This theorem is referenced by:  cbvrabv2  39311  smfsup  41020  smfinflem  41023  smfinf  41024