Theorem dfifp3

Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019)

	Ref	Expression
Assertion	dfifp3	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜑 ∨ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfifp2	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) )
2		pm4.64	⊢ ( ( ¬ 𝜑 → 𝜒 ) ↔ ( 𝜑 ∨ 𝜒 ) )
3	2	anbi2i	⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜑 ∨ 𝜒 ) ) )
4	1 3	bitri	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜑 ∨ 𝜒 ) ) )