Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  csbcomg Unicode version

Theorem csbcomg 3025
 Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
Assertion
Ref Expression
csbcomg
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()   (,)   (,)   (,)

Proof of Theorem csbcomg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2697 . 2
2 elex 2697 . 2
3 sbccom 2984 . . . . . 6
43a1i 9 . . . . 5
5 sbcel2g 3023 . . . . . . 7
65sbcbidv 2967 . . . . . 6
76adantl 275 . . . . 5
8 sbcel2g 3023 . . . . . . 7
98sbcbidv 2967 . . . . . 6
109adantr 274 . . . . 5
114, 7, 103bitr3d 217 . . . 4
12 sbcel2g 3023 . . . . 5
1312adantr 274 . . . 4
14 sbcel2g 3023 . . . . 5
1514adantl 275 . . . 4
1611, 13, 153bitr3d 217 . . 3
1716eqrdv 2137 . 2
181, 2, 17syl2an 287 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1331   wcel 1480  cvv 2686  wsbc 2909  csb 3003 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004 This theorem is referenced by:  ovmpos  5894
 Copyright terms: Public domain W3C validator