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Theorem biimpexp 31925
Description: A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.)
Assertion
Ref Expression
biimpexp (((𝜑𝜓) → 𝜒) ↔ ((𝜑𝜓) → ((𝜓𝜑) → 𝜒)))

Proof of Theorem biimpexp
StepHypRef Expression
1 dfbi2 663 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
21imbi1i 338 . 2 (((𝜑𝜓) → 𝜒) ↔ (((𝜑𝜓) ∧ (𝜓𝜑)) → 𝜒))
3 impexp 461 . 2 ((((𝜑𝜓) ∧ (𝜓𝜑)) → 𝜒) ↔ ((𝜑𝜓) → ((𝜓𝜑) → 𝜒)))
42, 3bitri 264 1 (((𝜑𝜓) → 𝜒) ↔ ((𝜑𝜓) → ((𝜓𝜑) → 𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383
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 197  df-an 385
This theorem is referenced by:  axextdfeq  32029
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