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Theorem dfifp7 1039
Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
Assertion
Ref Expression
dfifp7 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜒𝜑) → (𝜑𝜓)))

Proof of Theorem dfifp7
StepHypRef Expression
1 orcom 401 . 2 (((𝜑𝜓) ∨ ¬ (𝜒𝜑)) ↔ (¬ (𝜒𝜑) ∨ (𝜑𝜓)))
2 dfifp6 1038 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ ¬ (𝜒𝜑)))
3 imor 427 . 2 (((𝜒𝜑) → (𝜑𝜓)) ↔ (¬ (𝜒𝜑) ∨ (𝜑𝜓)))
41, 2, 33bitr4i 292 1 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜒𝜑) → (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  if-wif 1032
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-or 384  df-an 385  df-ifp 1033
This theorem is referenced by: (None)
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