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Theorem sbccom 3338
 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 3337 . . . 4
2 sbccomlem 3337 . . . . . . 7
32sbcbii 3322 . . . . . 6
4 sbccomlem 3337 . . . . . 6
53, 4bitri 253 . . . . 5
65sbcbii 3322 . . . 4
7 sbccomlem 3337 . . . . 5
87sbcbii 3322 . . . 4
91, 6, 83bitr3i 279 . . 3
10 sbcco 3289 . . 3
11 sbcco 3289 . . 3
129, 10, 113bitr3i 279 . 2
13 sbcco 3289 . . 3
1413sbcbii 3322 . 2
15 sbcco 3289 . . 3
1615sbcbii 3322 . 2
1712, 14, 163bitr3i 279 1
 Colors of variables: wff setvar class Syntax hints:   wb 188  wsbc 3266 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3046  df-sbc 3267 This theorem is referenced by:  csbcom  3782  csbab  3796  mpt2xopovel  6963  fi1uzind  12647  wrd2ind  12829  elmptrab  20835  sbccom2  32358  sbcrot3  35628  csbabgOLD  37205
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