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Theorem sbccom 3407
 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:   ,   ,   ,

Proof of Theorem sbccom
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbccomlem 3406 . . . 4
2 sbccomlem 3406 . . . . . . 7
32sbcbii 3387 . . . . . 6
4 sbccomlem 3406 . . . . . 6
53, 4bitri 249 . . . . 5
65sbcbii 3387 . . . 4
7 sbccomlem 3406 . . . . 5
87sbcbii 3387 . . . 4
91, 6, 83bitr3i 275 . . 3
10 sbcco 3350 . . 3
11 sbcco 3350 . . 3
129, 10, 113bitr3i 275 . 2
13 sbcco 3350 . . 3
1413sbcbii 3387 . 2
15 sbcco 3350 . . 3
1615sbcbii 3387 . 2
1712, 14, 163bitr3i 275 1
 Colors of variables: wff setvar class Syntax hints:  <->wb 184  [.wsbc 3327 This theorem is referenced by:  csbcom  3837  csbcomgOLD  3838  csbab  3855  csbabgOLD  3856  mpt2xopovel  6967  wrd2ind  12703  elmptrab  20328  sbccom2  30530  sbcrot3  30724 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-v 3111  df-sbc 3328
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