Theorem rabxfrd 4673

Description: Class builder membership after substituting an expression (containing ) for in the class expression . (Contributed by NM, 16-Jan-2012.)

Hypotheses

Ref	Expression
rabxfrd.1
rabxfrd.2
rabxfrd.3
rabxfrd.4
rabxfrd.5

Assertion

Ref	Expression
rabxfrd

Distinct variable groups:   ,   ,,   ,   ,   ,

Proof of Theorem rabxfrd

Step	Hyp	Ref	Expression
1		rabxfrd.3	. . . . . . . . . . 11
2	1	ex 434	. . . . . . . . . 10
3		ibibr 343	. . . . . . . . . 10
4	2, 3	sylib 196	. . . . . . . . 9
5	4	imp 429	. . . . . . . 8
6	5	anbi1d 704	. . . . . . 7
7		rabxfrd.4	. . . . . . . 8
8	7	elrab 3257	. . . . . . 7
9		rabid 3034	. . . . . . 7
10	6, 8, 9	3bitr4g 288	. . . . . 6
11	10	rabbidva 3100	. . . . 5
12	11	eleq2d 2527	. . . 4
13		rabxfrd.1	. . . . 5
14		nfcv 2619	. . . . 5
15		rabxfrd.2	. . . . . 6
16	15	nfel1 2635	. . . . 5
17		rabxfrd.5	. . . . . 6
18	17	eleq1d 2526	. . . . 5
19	13, 14, 16, 18	elrabf 3255	. . . 4
20		nfrab1 3038	. . . . . 6
21	13, 20	nfel 2632	. . . . 5
22		eleq1 2529	. . . . 5
23	13, 14, 21, 22	elrabf 3255	. . . 4
24	12, 19, 23	3bitr3g 287	. . 3
25		pm5.32 636	. . 3
26	24, 25	sylibr 212	. 2
27	26	imp 429	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  F/_wnfc 2605  {crab 2811

This theorem is referenced by:  rabxfr  4674  riotaxfrd  6288

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-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-ral 2812  df-rab 2816  df-v 3111