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Theorem sbccom 2827
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 2826 . . . 4  [.  ]. [.  ]. [.  ]. [.  ].  [.  ]. [.  ]. [.  ]. [.  ].
2 sbccomlem 2826 . . . . . . 7  [.  ]. [.  ].  [.  ]. [.  ].
32sbcbii 2812 . . . . . 6  [.  ]. [.  ]. [.  ].  [.  ]. [.  ]. [.  ].
4 sbccomlem 2826 . . . . . 6  [.  ]. [.  ]. [.  ].  [.  ]. [.  ]. [.  ].
53, 4bitri 173 . . . . 5  [.  ]. [.  ]. [.  ].  [.  ]. [.  ]. [.  ].
65sbcbii 2812 . . . 4  [.  ]. [.  ]. [.  ]. [.  ].  [.  ]. [.  ]. [.  ]. [.  ].
7 sbccomlem 2826 . . . . 5  [.  ]. [.  ]. [.  ].  [.  ]. [.  ]. [.  ].
87sbcbii 2812 . . . 4  [.  ]. [.  ]. [.  ]. [.  ].  [.  ]. [.  ]. [.  ]. [.  ].
91, 6, 83bitr3i 199 . . 3  [.  ]. [.  ]. [.  ]. [.  ].  [.  ]. [.  ]. [.  ]. [.  ].
10 sbcco 2779 . . 3  [.  ]. [.  ]. [.  ]. [.  ].  [.  ]. [.  ]. [.  ].
11 sbcco 2779 . . 3  [.  ]. [.  ]. [.  ]. [.  ].  [.  ]. [.  ]. [.  ].
129, 10, 113bitr3i 199 . 2  [.  ]. [.  ]. [.  ].  [.  ]. [.  ]. [.  ].
13 sbcco 2779 . . 3  [.  ]. [.  ].  [.  ].
1413sbcbii 2812 . 2  [.  ]. [.  ]. [.  ].  [.  ]. [.  ].
15 sbcco 2779 . . 3  [.  ]. [.  ].  [.  ].
1615sbcbii 2812 . 2  [.  ]. [.  ]. [.  ].  [.  ]. [.  ].
1712, 14, 163bitr3i 199 1  [.  ]. [.  ].  [.  ]. [.  ].
Colors of variables: wff set class
Syntax hints:   wb 98   [.wsbc 2758
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759
This theorem is referenced by:  csbcomg  2867  csbabg  2901  mpt2xopovel  5797
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