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

Theorem rabxfrd 4736
 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:   ,   ,,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()   (,)   (,)

Proof of Theorem rabxfrd
StepHypRef Expression
1 rabxfrd.3 . . . . . . . . . . 11
21ex 424 . . . . . . . . . 10
3 ibibr 333 . . . . . . . . . 10
42, 3sylib 189 . . . . . . . . 9
54imp 419 . . . . . . . 8
65anbi1d 686 . . . . . . 7
7 rabxfrd.4 . . . . . . . 8
87elrab 3084 . . . . . . 7
9 rabid 2876 . . . . . . 7
106, 8, 93bitr4g 280 . . . . . 6
1110rabbidva 2939 . . . . 5
1211eleq2d 2502 . . . 4
13 rabxfrd.1 . . . . 5
14 nfcv 2571 . . . . 5
15 rabxfrd.2 . . . . . 6
1615nfel1 2581 . . . . 5
17 rabxfrd.5 . . . . . 6
1817eleq1d 2501 . . . . 5
1913, 14, 16, 18elrabf 3083 . . . 4
20 nfrab1 2880 . . . . . 6
2113, 20nfel 2579 . . . . 5
22 eleq1 2495 . . . . 5
2313, 14, 21, 22elrabf 3083 . . . 4
2412, 19, 233bitr3g 279 . . 3
25 pm5.32 618 . . 3
2624, 25sylibr 204 . 2
2726imp 419 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wnfc 2558  crab 2701 This theorem is referenced by:  rabxfr  4737  riotaxfrd  6573 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950
 Copyright terms: Public domain W3C validator