Theorem dfifp4 1063

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 1062	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))
2		imor 849	. . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
3	2	anbi1i 623	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
4	1, 3	bitri 274	1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843  if-wif 1059

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 206  df-an 396  df-or 844  df-ifp 1060

This theorem is referenced by:  anifp  1068  ifpan123g  40964  ifpan23  40965  ifpdfor2  40966  ifpdfor  40970  ifpim1  40974  ifpnot  40975  ifpid2  40976  ifpim2  40977  ifpnot23  40983  ifpidg  40996  ifpim123g  41005  ifpimim  41014