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

Theorem csbopabg 4275
 Description: Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
csbopabg
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,,)   ()   (,,)

Proof of Theorem csbopabg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3246 . . 3
2 dfsbcq2 3156 . . . 4
32opabbidv 4263 . . 3
41, 3eqeq12d 2449 . 2
5 vex 2951 . . 3
6 nfs1v 2181 . . . 4
76nfopab 4265 . . 3
8 sbequ12 1944 . . . 4
98opabbidv 4263 . . 3
105, 7, 9csbief 3284 . 2
114, 10vtoclg 3003 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652  wsb 1658   wcel 1725  wsbc 3153  csb 3243  copab 4257 This theorem is referenced by:  csbcnvg  24029 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-3an 938  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-v 2950  df-sbc 3154  df-csb 3244  df-opab 4259
 Copyright terms: Public domain W3C validator