Theorem dfifp6

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

	Ref	Expression
Assertion	dfifp6	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ¬ ( 𝜒 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-ifp	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜒 ) ) )
2		ancom	⊢ ( ( ¬ 𝜑 ∧ 𝜒 ) ↔ ( 𝜒 ∧ ¬ 𝜑 ) )
3		annim	⊢ ( ( 𝜒 ∧ ¬ 𝜑 ) ↔ ¬ ( 𝜒 → 𝜑 ) )
4	2 3	bitri	⊢ ( ( ¬ 𝜑 ∧ 𝜒 ) ↔ ¬ ( 𝜒 → 𝜑 ) )
5	4	orbi2i	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ¬ ( 𝜒 → 𝜑 ) ) )
6	1 5	bitri	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ¬ ( 𝜒 → 𝜑 ) ) )