Theorem ifpdfor

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

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

Proof

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