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Theorem csbcomg 2930
 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 2611 . 2
2 elex 2611 . 2
3 sbccom 2890 . . . . . 6
43a1i 9 . . . . 5
5 sbcel2g 2928 . . . . . . 7
65sbcbidv 2873 . . . . . 6
76adantl 271 . . . . 5
8 sbcel2g 2928 . . . . . . 7
98sbcbidv 2873 . . . . . 6
109adantr 270 . . . . 5
114, 7, 103bitr3d 216 . . . 4
12 sbcel2g 2928 . . . . 5
1312adantr 270 . . . 4
14 sbcel2g 2928 . . . . 5
1514adantl 271 . . . 4
1611, 13, 153bitr3d 216 . . 3
1716eqrdv 2080 . 2
181, 2, 17syl2an 283 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wb 103   wceq 1285   wcel 1434  cvv 2602  wsbc 2816  csb 2909 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817  df-csb 2910 This theorem is referenced by:  ovmpt2s  5655
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