Theorem dfifp4 1082

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 1081	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))
2		imor 867	. 2 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
3	1, 2	bianbi 639	1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079

This theorem is used by:  anifp  1088  ifpan123g  44606  ifpan23  44607  ifpdfor2  44608  ifpdfor  44612  ifpim1  44616  ifpnot  44617  ifpid2  44618  ifpim2  44619  ifpnot23  44625  ifpidg  44638  ifpim123g  44647  ifpimim  44656