Theorem ifpdfor2 39846

Description: Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.)

Assertion

Ref	Expression
ifpdfor2	⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, 𝜑, 𝜓))

Proof of Theorem ifpdfor2

Step	Hyp	Ref	Expression
1		pm2.1 893	. . 3 ⊢ (¬ 𝜑 ∨ 𝜑)
2	1	biantrur 533	. 2 ⊢ ((𝜑 ∨ 𝜓) ↔ ((¬ 𝜑 ∨ 𝜑) ∧ (𝜑 ∨ 𝜓)))
3		dfifp4 1061	. 2 ⊢ (if-(𝜑, 𝜑, 𝜓) ↔ ((¬ 𝜑 ∨ 𝜑) ∧ (𝜑 ∨ 𝜓)))
4	2, 3	bitr4i 280	1 ⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, 𝜑, 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843  if-wif 1057

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 209  df-an 399  df-or 844  df-ifp 1058

This theorem is referenced by:  ifporcor  39847