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Theorem sbc3an 3847
Description: Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbc3an ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

Proof of Theorem sbc3an
StepHypRef Expression
1 df-3an 1089 . . . 4 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
21sbcbii 3837 . . 3 ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ [𝐴 / 𝑥]((𝜑𝜓) ∧ 𝜒))
3 sbcan 3829 . . 3 ([𝐴 / 𝑥]((𝜑𝜓) ∧ 𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∧ [𝐴 / 𝑥]𝜒))
4 sbcan 3829 . . . 4 ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
54anbi1i 624 . . 3 (([𝐴 / 𝑥](𝜑𝜓) ∧ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∧ [𝐴 / 𝑥]𝜒))
62, 3, 53bitri 296 . 2 ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∧ [𝐴 / 𝑥]𝜒))
7 df-3an 1089 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∧ [𝐴 / 𝑥]𝜒))
86, 7bitr4i 277 1 ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1087  [wsbc 3777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-sbc 3778
This theorem is referenced by:  csbfrecsg  8268  bnj156  33734  bnj206  33737  bnj976  33783  bnj121  33876  bnj130  33880  bnj581  33914  bnj1040  33978  topdifinffinlem  36223  rdgeqoa  36246  cdlemkid3N  39799  cdlemkid4  39800  minregex  42275
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