Theorem sbcabel 3416

Description: Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)

Hypothesis

Ref	Expression
sbcabel.1

Assertion

Ref	Expression
sbcabel

Distinct variable groups:   ,   ,

Proof of Theorem sbcabel

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3118	. 2
2		sbcex2 3381	. . . 4
3		sbcan 3370	. . . . . 6
4		sbcal 3379	. . . . . . . . 9
5		sbcbig 3374	. . . . . . . . . . 11
6		sbcg 3401	. . . . . . . . . . . 12
7	6	bibi1d 319	. . . . . . . . . . 11
8	5, 7	bitrd 253	. . . . . . . . . 10
9	8	albidv 1713	. . . . . . . . 9
10	4, 9	syl5bb 257	. . . . . . . 8
11		abeq2 2581	. . . . . . . . 9
12	11	sbcbii 3387	. . . . . . . 8
13		abeq2 2581	. . . . . . . 8
14	10, 12, 13	3bitr4g 288	. . . . . . 7
15		sbcabel.1	. . . . . . . . 9
16	15	nfcri 2612	. . . . . . . 8
17	16	sbcgf 3399	. . . . . . 7
18	14, 17	anbi12d 710	. . . . . 6
19	3, 18	syl5bb 257	. . . . 5
20	19	exbidv 1714	. . . 4
21	2, 20	syl5bb 257	. . 3
22		df-clel 2452	. . . 4
23	22	sbcbii 3387	. . 3
24		df-clel 2452	. . 3
25	21, 23, 24	3bitr4g 288	. 2
26	1, 25	syl 16	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  F/_wnfc 2605  cvv 3109  [.wsbc 3327

This theorem is referenced by:  csbexg  4584  csbexgOLD  4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328