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

Theorem csbdm 5202
 Description: Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.) (Revised by NM, 24-Aug-2018.)
Assertion
Ref Expression
csbdm

Proof of Theorem csbdm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbab 3855 . . 3
2 sbcex2 3381 . . . . 5
3 sbcel2 3831 . . . . . 6
43exbii 1667 . . . . 5
52, 4bitri 249 . . . 4
65abbii 2591 . . 3
71, 6eqtri 2486 . 2
8 dfdm3 5195 . . 3
98csbeq2i 3836 . 2
10 dfdm3 5195 . 2
117, 9, 103eqtr4i 2496 1
 Colors of variables: wff setvar class Syntax hints:  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  [.wsbc 3327  [_csb 3434  <.cop 4035  domcdm 5004 This theorem is referenced by:  sbcfng  5733 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-or 370  df-an 371  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-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-br 4453  df-dm 5014
 Copyright terms: Public domain W3C validator