Theorem ifpbi23d

Description: Equivalence deduction for conditional operator for propositions. Convenience theorem for a frequent case. (Contributed by Wolf Lammen, 28-Apr-2024)

	Ref	Expression
Hypotheses	ifpbi23d.1	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜂 ) )
	ifpbi23d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜁 ) )
Assertion	ifpbi23d	⊢ ( 𝜑 → ( if- ( 𝜓 , 𝜒 , 𝜃 ) ↔ if- ( 𝜓 , 𝜂 , 𝜁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ifpbi23d.1	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜂 ) )
2		ifpbi23d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜁 ) )
3		biidd	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜓 ) )
4	3 1 2	ifpbi123d	⊢ ( 𝜑 → ( if- ( 𝜓 , 𝜒 , 𝜃 ) ↔ if- ( 𝜓 , 𝜂 , 𝜁 ) ) )