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Theorem sbc3an 3835
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 1081 . . . 4 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
21sbcbii 3826 . . 3 ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ [𝐴 / 𝑥]((𝜑𝜓) ∧ 𝜒))
3 sbcan 3818 . . 3 ([𝐴 / 𝑥]((𝜑𝜓) ∧ 𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∧ [𝐴 / 𝑥]𝜒))
4 sbcan 3818 . . . 4 ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
54anbi1i 623 . . 3 (([𝐴 / 𝑥](𝜑𝜓) ∧ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∧ [𝐴 / 𝑥]𝜒))
62, 3, 53bitri 298 . 2 ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∧ [𝐴 / 𝑥]𝜒))
7 df-3an 1081 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∧ [𝐴 / 𝑥]𝜒))
86, 7bitr4i 279 1 ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1079  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-v 3494  df-sbc 3770
This theorem is referenced by:  bnj156  31897  bnj206  31900  bnj976  31948  bnj121  32041  bnj130  32045  bnj581  32079  bnj1040  32141  csbwrecsg  34490  topdifinffinlem  34510  rdgeqoa  34533  cdlemkid3N  37949  cdlemkid4  37950
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