Theorem ifbieq2d

Description: Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011)

	Ref	Expression
Hypotheses	ifbieq2d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	ifbieq2d.2	$$⊢\phi \to A=B$$
Assertion	ifbieq2d	$$⊢\phi \to if\left(\psi ,C,A\right)=if\left(\chi ,C,B\right)$$

Proof

Step	Hyp	Ref	Expression
1		ifbieq2d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		ifbieq2d.2	$$⊢\phi \to A=B$$
3	1	ifbid	$$⊢\phi \to if\left(\psi ,C,A\right)=if\left(\chi ,C,A\right)$$
4	2	ifeq2d	$$⊢\phi \to if\left(\chi ,C,A\right)=if\left(\chi ,C,B\right)$$
5	3 4	eqtrd	$$⊢\phi \to if\left(\psi ,C,A\right)=if\left(\chi ,C,B\right)$$