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Theorem sbcani 35504
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 3795 . 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 629 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ↔ (𝜒𝜂))
51, 4bitri 278 1 ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜒𝜂))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  [wsbc 3747
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-v 3471  df-sbc 3748
This theorem is referenced by: (None)
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