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Theorem sbcco 2785
 Description: A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbcco
Distinct variable group:   ,
Allowed substitution hints:   ()   (,)

Proof of Theorem sbcco
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbcex 2772 . 2
2 sbcex 2772 . 2
3 dfsbcq 2766 . . 3
4 dfsbcq 2766 . . 3
5 sbsbc 2768 . . . . . 6
65sbbii 1648 . . . . 5
7 nfv 1421 . . . . . 6
87sbco2 1839 . . . . 5
9 sbsbc 2768 . . . . 5
106, 8, 93bitr3ri 200 . . . 4
11 sbsbc 2768 . . . 4
1210, 11bitri 173 . . 3
133, 4, 12vtoclbg 2614 . 2
141, 2, 13pm5.21nii 620 1
 Colors of variables: wff set class Syntax hints:   wb 98   wcel 1393  wsb 1645  cvv 2557  wsbc 2764 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765 This theorem is referenced by:  sbc7  2790  sbccom  2833  sbcralt  2834  sbcrext  2835  csbco  2861
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