Theorem dfifp5 1011

Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)

Assertion

Ref	Expression
dfifp5	⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜑 → 𝜒)))

Proof of Theorem dfifp5

Step	Hyp	Ref	Expression
1		dfifp2 1008	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
2		imor 427	. . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
3	2	anbi1i 727	. 2 ⊢ (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜑 → 𝜒)))
4	1, 3	bitri 263	1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜑 → 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383  if-wif 1006

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 196  df-or 384  df-an 385  df-ifp 1007

This theorem is referenced by: (None)