Theorem dfifp7

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

	Ref	Expression
Assertion	dfifp7	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜒 → 𝜑 ) → ( 𝜑 ∧ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		orcom	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ¬ ( 𝜒 → 𝜑 ) ) ↔ ( ¬ ( 𝜒 → 𝜑 ) ∨ ( 𝜑 ∧ 𝜓 ) ) )
2		dfifp6	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ¬ ( 𝜒 → 𝜑 ) ) )
3		imor	⊢ ( ( ( 𝜒 → 𝜑 ) → ( 𝜑 ∧ 𝜓 ) ) ↔ ( ¬ ( 𝜒 → 𝜑 ) ∨ ( 𝜑 ∧ 𝜓 ) ) )
4	1 2 3	3bitr4i	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜒 → 𝜑 ) → ( 𝜑 ∧ 𝜓 ) ) )