Theorem dfifp4 1073

Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)

Assertion

Ref	Expression
dfifp4	⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

Proof of Theorem dfifp4

Step	Hyp	Ref	Expression
1		dfifp3 1072	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))
2		imor 860	. 2 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
3	1, 2	bianbi 634	1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∨ wo 854  if-wif 1069

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 209  df-an 398  df-or 855  df-ifp 1070

This theorem is referenced by:  anifp  1078  ifpan123g  43918  ifpan23  43919  ifpdfor2  43920  ifpdfor  43924  ifpim1  43928  ifpnot  43929  ifpid2  43930  ifpim2  43931  ifpnot23  43937  ifpidg  43950  ifpim123g  43959  ifpimim  43968