Theorem ifpdfor 44232

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

Assertion

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

Proof of Theorem ifpdfor

Step	Hyp	Ref	Expression
1		tru 1574	. . . 4 ⊢ ⊤
2	1	olci 880	. . 3 ⊢ (¬ 𝜑 ∨ ⊤)
3	2	biantrur 540	. 2 ⊢ ((𝜑 ∨ 𝜓) ↔ ((¬ 𝜑 ∨ ⊤) ∧ (𝜑 ∨ 𝜓)))
4		dfifp4 1082	. 2 ⊢ (if-(𝜑, ⊤, 𝜓) ↔ ((¬ 𝜑 ∨ ⊤) ∧ (𝜑 ∨ 𝜓)))
5	3, 4	bitr4i 281	1 ⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, ⊤, 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861  if-wif 1078  ⊤wtru 1571

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-tru 1573

This theorem is used by:  ifpdfxor  44254