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Theorem sbcco 2934
 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 2921 . 2
2 sbcex 2921 . 2
3 dfsbcq 2915 . . 3
4 dfsbcq 2915 . . 3
5 sbsbc 2917 . . . . . 6
65sbbii 1739 . . . . 5
7 nfv 1509 . . . . . 6
87sbco2 1939 . . . . 5
9 sbsbc 2917 . . . . 5
106, 8, 93bitr3ri 210 . . . 4
11 sbsbc 2917 . . . 4
1210, 11bitri 183 . . 3
133, 4, 12vtoclbg 2750 . 2
141, 2, 13pm5.21nii 694 1
 Colors of variables: wff set class Syntax hints:   wb 104   wcel 1481  wsb 1736  cvv 2689  wsbc 2913 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2914 This theorem is referenced by:  sbc7  2939  sbccom  2988  sbcralt  2989  sbcrext  2990  csbco  3017
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