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

Theorem csbcomgOLD 3838
 Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) Obsolete as of 18-Aug-2018. Use csbcom 3837 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbcomgOLD
Distinct variable groups:   ,   ,   ,

Proof of Theorem csbcomgOLD
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3118 . 2
2 elex 3118 . 2
3 sbccom 3407 . . . . . 6
43a1i 11 . . . . 5
5 sbcel2gOLD 3832 . . . . . . 7
65sbcbidv 3386 . . . . . 6
76adantl 466 . . . . 5
8 sbcel2gOLD 3832 . . . . . . 7
98sbcbidv 3386 . . . . . 6
109adantr 465 . . . . 5
114, 7, 103bitr3d 283 . . . 4
12 sbcel2gOLD 3832 . . . . 5
1312adantr 465 . . . 4
14 sbcel2gOLD 3832 . . . . 5
1514adantl 466 . . . 4
1611, 13, 153bitr3d 283 . . 3
1716eqrdv 2454 . 2
181, 2, 17syl2an 477 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818   cvv 3109  [.wsbc 3327  [_csb 3434 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-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
 Copyright terms: Public domain W3C validator