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Theorem sbcang 2800
Description: Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.)
Assertion
Ref Expression
sbcang  V  [.  ].  [.  ].  [.  ].

Proof of Theorem sbcang
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2761 . 2 
[.  ].
2 dfsbcq2 2761 . . 3  [.  ].
3 dfsbcq2 2761 . . 3  [.  ].
42, 3anbi12d 442 . 2  [.  ].  [.  ].
5 sban 1826 . 2
61, 4, 5vtoclbg 2608 1  V  [.  ].  [.  ].  [.  ].
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390  wsb 1642   [.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:  sbcabel  2833  csbunig  3579  csbxpg  4364
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