Theorem ifpimpda

Description: Separation of the values of the conditional operator for propositions. (Contributed by AV, 30-Dec-2020) (Proof shortened by Wolf Lammen, 27-Feb-2021)

	Ref	Expression
Hypotheses	ifpimpda.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
	ifpimpda.2	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝜃 )
Assertion	ifpimpda	⊢ ( 𝜑 → if- ( 𝜓 , 𝜒 , 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		ifpimpda.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2		ifpimpda.2	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝜃 )
3	1	ex	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
4	2	ex	⊢ ( 𝜑 → ( ¬ 𝜓 → 𝜃 ) )
5		dfifp2	⊢ ( if- ( 𝜓 , 𝜒 , 𝜃 ) ↔ ( ( 𝜓 → 𝜒 ) ∧ ( ¬ 𝜓 → 𝜃 ) ) )
6	3 4 5	sylanbrc	⊢ ( 𝜑 → if- ( 𝜓 , 𝜒 , 𝜃 ) )