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Theorem sbc3an 3461
 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 1033 . . . 4 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
21sbcbii 3458 . . 3 ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ [𝐴 / 𝑥]((𝜑𝜓) ∧ 𝜒))
3 sbcan 3445 . . 3 ([𝐴 / 𝑥]((𝜑𝜓) ∧ 𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∧ [𝐴 / 𝑥]𝜒))
4 sbcan 3445 . . . 4 ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
54anbi1i 727 . . 3 (([𝐴 / 𝑥](𝜑𝜓) ∧ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∧ [𝐴 / 𝑥]𝜒))
62, 3, 53bitri 285 . 2 ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∧ [𝐴 / 𝑥]𝜒))
7 df-3an 1033 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∧ [𝐴 / 𝑥]𝜒))
86, 7bitr4i 266 1 ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403 This theorem is referenced by:  bnj156  30050  bnj206  30053  bnj976  30102  bnj121  30194  bnj130  30198  bnj581  30232  bnj1040  30294  csbwrecsg  32349  topdifinffinlem  32371  rdgeqoa  32394  cdlemkid3N  35239  cdlemkid4  35240
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