Theorem ifbieq1d

Description: Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018)

	Ref	Expression
Hypotheses	ifbieq1d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	ifbieq1d.2	$$⊢\phi \to A=B$$
Assertion	ifbieq1d	$$⊢\phi \to if\left(\psi ,A,C\right)=if\left(\chi ,B,C\right)$$

Proof

Step	Hyp	Ref	Expression
1		ifbieq1d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		ifbieq1d.2	$$⊢\phi \to A=B$$
3	1	ifbid	$$⊢\phi \to if\left(\psi ,A,C\right)=if\left(\chi ,A,C\right)$$
4	2	ifeq1d	$$⊢\phi \to if\left(\chi ,A,C\right)=if\left(\chi ,B,C\right)$$
5	3 4	eqtrd	$$⊢\phi \to if\left(\psi ,A,C\right)=if\left(\chi ,B,C\right)$$