Theorem anifp

Description: The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor . (Contributed by BJ, 30-Sep-2019)

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

Proof

Step	Hyp	Ref	Expression
1		olc	⊢ ( 𝜓 → ( ¬ 𝜑 ∨ 𝜓 ) )
2		olc	⊢ ( 𝜒 → ( 𝜑 ∨ 𝜒 ) )
3	1 2	anim12i	⊢ ( ( 𝜓 ∧ 𝜒 ) → ( ( ¬ 𝜑 ∨ 𝜓 ) ∧ ( 𝜑 ∨ 𝜒 ) ) )
4		dfifp4	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( ¬ 𝜑 ∨ 𝜓 ) ∧ ( 𝜑 ∨ 𝜒 ) ) )
5	3 4	sylibr	⊢ ( ( 𝜓 ∧ 𝜒 ) → if- ( 𝜑 , 𝜓 , 𝜒 ) )