Theorem dfifp7 1013

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

Assertion

Ref	Expression
dfifp7	⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜒 → 𝜑) → (𝜑 ∧ 𝜓)))

Proof of Theorem dfifp7

Step	Hyp	Ref	Expression
1		orcom 401	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ ¬ (𝜒 → 𝜑)) ↔ (¬ (𝜒 → 𝜑) ∨ (𝜑 ∧ 𝜓)))
2		dfifp6 1012	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜒 → 𝜑)))
3		imor 427	. 2 ⊢ (((𝜒 → 𝜑) → (𝜑 ∧ 𝜓)) ↔ (¬ (𝜒 → 𝜑) ∨ (𝜑 ∧ 𝜓)))
4	1, 2, 3	3bitr4i 291	1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜒 → 𝜑) → (𝜑 ∧ 𝜓)))

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)