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

Theorem csbopab 4784
 Description: Move substitution into a class abstraction. Version of csbopabgALT 4785 without a sethood antecedent but depending on more axioms. (Contributed by NM, 6-Aug-2007.) (Revised by NM, 23-Aug-2018.)
Assertion
Ref Expression
csbopab
Distinct variable groups:   ,,   ,,

Proof of Theorem csbopab
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3437 . . . 4
2 dfsbcq2 3330 . . . . 5
32opabbidv 4515 . . . 4
41, 3eqeq12d 2479 . . 3
5 vex 3112 . . . 4
6 nfs1v 2181 . . . . 5
76nfopab 4517 . . . 4
8 sbequ12 1992 . . . . 5
98opabbidv 4515 . . . 4
105, 7, 9csbief 3459 . . 3
114, 10vtoclg 3167 . 2
12 csbprc 3821 . . 3
13 sbcex 3337 . . . . . . 7
1413con3i 135 . . . . . 6
1514nexdv 1884 . . . . 5
1615nexdv 1884 . . . 4
17 opabn0 4783 . . . . 5
1817necon1bbii 2721 . . . 4
1916, 18sylib 196 . . 3
2012, 19eqtr4d 2501 . 2
2111, 20pm2.61i 164 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  =wceq 1395  E.wex 1612  [wsb 1739  e.wcel 1818   cvv 3109  [.wsbc 3327  [_csb 3434   c0 3784  {copab 4509 This theorem is referenced by:  csbmpt12  4786  csbcnv  5191 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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-opab 4511
 Copyright terms: Public domain W3C validator