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Theorem sbcani 33080
 Description: Distribution of class substitution over conjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
Hypotheses
Ref Expression
sbcani.1 ([𝐴 / 𝑥]𝜑𝜒)
sbcani.2 ([𝐴 / 𝑥]𝜓𝜂)
Assertion
Ref Expression
sbcani ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜒𝜂))

Proof of Theorem sbcani
StepHypRef Expression
1 sbcan 3445 . 2 ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
2 sbcani.1 . . 3 ([𝐴 / 𝑥]𝜑𝜒)
3 sbcani.2 . . 3 ([𝐴 / 𝑥]𝜓𝜂)
42, 3anbi12i 729 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ↔ (𝜒𝜂))
51, 4bitri 263 1 ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜒𝜂))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  [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-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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: (None)
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