Theorem ifpim2 40977

Description: Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)

Assertion

Ref	Expression
ifpim2	⊢ ((𝜑 → 𝜓) ↔ if-(𝜓, ⊤, ¬ 𝜑))

Proof of Theorem ifpim2

Step	Hyp	Ref	Expression
1		tru 1543	. . . 4 ⊢ ⊤
2	1	olci 862	. . 3 ⊢ (¬ 𝜓 ∨ ⊤)
3	2	biantrur 530	. 2 ⊢ ((𝜓 ∨ ¬ 𝜑) ↔ ((¬ 𝜓 ∨ ⊤) ∧ (𝜓 ∨ ¬ 𝜑)))
4		imor 849	. . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
5		orcom 866	. . 3 ⊢ ((¬ 𝜑 ∨ 𝜓) ↔ (𝜓 ∨ ¬ 𝜑))
6	4, 5	bitri 274	. 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜓 ∨ ¬ 𝜑))
7		dfifp4 1063	. 2 ⊢ (if-(𝜓, ⊤, ¬ 𝜑) ↔ ((¬ 𝜓 ∨ ⊤) ∧ (𝜓 ∨ ¬ 𝜑)))
8	3, 6, 7	3bitr4i 302	1 ⊢ ((𝜑 → 𝜓) ↔ if-(𝜓, ⊤, ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843  if-wif 1059  ⊤wtru 1540

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  df-tru 1542

This theorem is referenced by: (None)