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Theorem sbccom 3117
 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 3116 . . . 4
2 sbccomlem 3116 . . . . . . 7
32sbcbii 3101 . . . . . 6
4 sbccomlem 3116 . . . . . 6
53, 4bitri 240 . . . . 5
65sbcbii 3101 . . . 4
7 sbccomlem 3116 . . . . 5
87sbcbii 3101 . . . 4
91, 6, 83bitr3i 266 . . 3
10 sbcco 3068 . . 3
11 sbcco 3068 . . 3
129, 10, 113bitr3i 266 . 2
13 sbcco 3068 . . 3
1413sbcbii 3101 . 2
15 sbcco 3068 . . 3
1615sbcbii 3101 . 2
1712, 14, 163bitr3i 266 1
 Colors of variables: wff setvar class Syntax hints:   wb 176  wsbc 3046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047 This theorem is referenced by:  csbcomg  3159  csbabg  3197  cnvopab  5030  eqerlem  5960
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