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Theorem sbc3an 3860
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 sbcan 3843 . . 3 ([𝐴 / 𝑥]((𝜑𝜓) ∧ 𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∧ [𝐴 / 𝑥]𝜒))
2 sbcan 3843 . . 3 ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
31, 2bianbi 627 . 2 ([𝐴 / 𝑥]((𝜑𝜓) ∧ 𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∧ [𝐴 / 𝑥]𝜒))
4 df-3an 1088 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
54sbcbii 3851 . 2 ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ [𝐴 / 𝑥]((𝜑𝜓) ∧ 𝜒))
6 df-3an 1088 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∧ [𝐴 / 𝑥]𝜒))
73, 5, 63bitr4i 303 1 ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086  [wsbc 3790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-sbc 3791
This theorem is referenced by:  csbfrecsg  8307  bnj156  34720  bnj206  34723  bnj976  34769  bnj121  34862  bnj130  34866  bnj581  34900  bnj1040  34964  topdifinffinlem  37329  rdgeqoa  37352  cdlemkid3N  40915  cdlemkid4  40916  minregex  43523
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