Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabxfrd Unicode version

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
StepHypRef Expression
1 rabxfrd.3 . . . . . . . . . . 11
21ex 434 . . . . . . . . . 10
3 ibibr 343 . . . . . . . . . 10
42, 3sylib 196 . . . . . . . . 9
54imp 429 . . . . . . . 8
65anbi1d 704 . . . . . . 7
7 rabxfrd.4 . . . . . . . 8
87elrab 3257 . . . . . . 7
9 rabid 3034 . . . . . . 7
106, 8, 93bitr4g 288 . . . . . 6
1110rabbidva 3100 . . . . 5
1211eleq2d 2527 . . . 4
13 rabxfrd.1 . . . . 5
14 nfcv 2619 . . . . 5
15 rabxfrd.2 . . . . . 6
1615nfel1 2635 . . . . 5
17 rabxfrd.5 . . . . . 6
1817eleq1d 2526 . . . . 5
1913, 14, 16, 18elrabf 3255 . . . 4
20 nfrab1 3038 . . . . . 6
2113, 20nfel 2632 . . . . 5
22 eleq1 2529 . . . . 5
2313, 14, 21, 22elrabf 3255 . . . 4
2412, 19, 233bitr3g 287 . . 3
25 pm5.32 636 . . 3
2624, 25sylibr 212 . 2
2726imp 429 1
 Colors of variables: wff setvar class 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
 Copyright terms: Public domain W3C validator