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Theorem sbcan 2799
Description: Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
Assertion
Ref Expression
sbcan  [.  ].  [.  ].  [.  ].

Proof of Theorem sbcan
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbcex 2766 . 2  [.  ].  _V
2 sbcex 2766 . . 3  [.  ].  _V
32adantl 262 . 2 
[.  ].  [.  ].  _V
4 dfsbcq2 2761 . . 3 
[.  ].
5 dfsbcq2 2761 . . . 4  [.  ].
6 dfsbcq2 2761 . . . 4  [.  ].
75, 6anbi12d 442 . . 3  [.  ].  [.  ].
8 sban 1826 . . 3
94, 7, 8vtoclbg 2608 . 2  _V  [.  ].  [.  ].  [.  ].
101, 3, 9pm5.21nii 619 1  [.  ].  [.  ].  [.  ].
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98   wceq 1242   wcel 1390  wsb 1642   _Vcvv 2551   [.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-bnd 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:  difopab  4412  sbcfung  4868  sbcfng  4987  sbcfg  4988
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