Theorem dfifp5

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

	Ref	Expression
Assertion	dfifp5	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( ¬ 𝜑 ∨ 𝜓 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfifp2	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) )
2		imor	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( ¬ 𝜑 ∨ 𝜓 ) )
3	2	anbi1i	⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) ↔ ( ( ¬ 𝜑 ∨ 𝜓 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) )
4	1 3	bitri	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( ¬ 𝜑 ∨ 𝜓 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) )