Theorem cdeqim

Description: Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016)

	Ref	Expression
Hypotheses	cdeqnot.1	$$⊢CondEq\left(x=y\to \left(\phi \leftrightarrow \psi \right)\right)$$
	cdeqim.1	$$⊢CondEq\left(x=y\to \left(\chi \leftrightarrow \theta \right)\right)$$
Assertion	cdeqim	$$⊢CondEq\left(x=y\to \left(\left(\phi \to \chi \right)\leftrightarrow \left(\psi \to \theta \right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdeqnot.1	$$⊢CondEq\left(x=y\to \left(\phi \leftrightarrow \psi \right)\right)$$
2		cdeqim.1	$$⊢CondEq\left(x=y\to \left(\chi \leftrightarrow \theta \right)\right)$$
3	1	cdeqri	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4	2	cdeqri	$$⊢x=y\to \left(\chi \leftrightarrow \theta \right)$$
5	3 4	imbi12d	$$⊢x=y\to \left(\left(\phi \to \chi \right)\leftrightarrow \left(\psi \to \theta \right)\right)$$
6	5	cdeqi	$$⊢CondEq\left(x=y\to \left(\left(\phi \to \chi \right)\leftrightarrow \left(\psi \to \theta \right)\right)\right)$$