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	$$⊢\phi \to \left(\chi \leftrightarrow \eta \right)$$
	ifpbi23d.2	$$⊢\phi \to \left(\theta \leftrightarrow \zeta \right)$$
Assertion	ifpbi23d	$$⊢\phi \to \left(if-\left(\psi ,\chi ,\theta \right)\leftrightarrow if-\left(\psi ,\eta ,\zeta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ifpbi23d.1	$$⊢\phi \to \left(\chi \leftrightarrow \eta \right)$$
2		ifpbi23d.2	$$⊢\phi \to \left(\theta \leftrightarrow \zeta \right)$$
3		biidd	$$⊢\phi \to \left(\psi \leftrightarrow \psi \right)$$
4	3 1 2	ifpbi123d	$$⊢\phi \to \left(if-\left(\psi ,\chi ,\theta \right)\leftrightarrow if-\left(\psi ,\eta ,\zeta \right)\right)$$