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Theorem sbccom 2833
 Description: Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
Assertion
Ref Expression
sbccom
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   (,)   ()   ()

Proof of Theorem sbccom
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbccomlem 2832 . . . 4
2 sbccomlem 2832 . . . . . . 7
32sbcbii 2818 . . . . . 6
4 sbccomlem 2832 . . . . . 6
53, 4bitri 173 . . . . 5
65sbcbii 2818 . . . 4
7 sbccomlem 2832 . . . . 5
87sbcbii 2818 . . . 4
91, 6, 83bitr3i 199 . . 3
10 sbcco 2785 . . 3
11 sbcco 2785 . . 3
129, 10, 113bitr3i 199 . 2
13 sbcco 2785 . . 3
1413sbcbii 2818 . 2
15 sbcco 2785 . . 3
1615sbcbii 2818 . 2
1712, 14, 163bitr3i 199 1
 Colors of variables: wff set class Syntax hints:   wb 98  wsbc 2764 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765 This theorem is referenced by:  csbcomg  2873  csbabg  2907  mpt2xopovel  5856
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