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Theorem pm5.32dav 45657
Description: Distribution of implication over biconditional (deduction form). Variant of pm5.32da 582. (Contributed by Zhi Wang, 30-Aug-2024.)
Hypothesis
Ref Expression
pm5.32dav.1 ((𝜑𝜓) → (𝜒𝜃))
Assertion
Ref Expression
pm5.32dav (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜓)))

Proof of Theorem pm5.32dav
StepHypRef Expression
1 pm5.32dav.1 . . 3 ((𝜑𝜓) → (𝜒𝜃))
21pm5.32da 582 . 2 (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
3 ancom 464 . 2 ((𝜓𝜒) ↔ (𝜒𝜓))
4 ancom 464 . 2 ((𝜓𝜃) ↔ (𝜃𝜓))
52, 3, 43bitr3g 316 1 (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 210  df-an 400
This theorem is referenced by: (None)
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