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Theorem sbccom 3233
 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 3232 . . . 4
2 sbccomlem 3232 . . . . . . 7
32sbcbii 3217 . . . . . 6
4 sbccomlem 3232 . . . . . 6
53, 4bitri 242 . . . . 5
65sbcbii 3217 . . . 4
7 sbccomlem 3232 . . . . 5
87sbcbii 3217 . . . 4
91, 6, 83bitr3i 268 . . 3
10 sbcco 3184 . . 3
11 sbcco 3184 . . 3
129, 10, 113bitr3i 268 . 2
13 sbcco 3184 . . 3
1413sbcbii 3217 . 2
15 sbcco 3184 . . 3
1615sbcbii 3217 . 2
1712, 14, 163bitr3i 268 1
 Colors of variables: wff set class Syntax hints:   wb 178  wsbc 3162 This theorem is referenced by:  csbcomg  3275  csbabg  3311  mpt2xopovel  6472  elmptrab  17860  sbcrot3  26848 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-sbc 3163
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