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Theorem pm5.32rd 432
Description: Distribution of implication over biconditional (deduction rule). (Contributed by NM, 25-Dec-2004.)
Hypothesis
Ref Expression
pm5.32d.1 (𝜑 → (𝜓 → (𝜒𝜃)))
Assertion
Ref Expression
pm5.32rd (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜓)))

Proof of Theorem pm5.32rd
StepHypRef Expression
1 pm5.32d.1 . . 3 (𝜑 → (𝜓 → (𝜒𝜃)))
21pm5.32d 431 . 2 (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
3 ancom 257 . 2 ((𝜒𝜓) ↔ (𝜓𝜒))
4 ancom 257 . 2 ((𝜃𝜓) ↔ (𝜓𝜃))
52, 3, 43bitr4g 216 1 (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  anbi1d  446  pm5.71dc  879  1idprl  6745  1idpru  6746
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