Theorem cdeqim

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

	Ref	Expression
Hypotheses	cdeqnot.1	⊢ CondEq ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
	cdeqim.1	⊢ CondEq ( 𝑥 = 𝑦 → ( 𝜒 ↔ 𝜃 ) )
Assertion	cdeqim	⊢ CondEq ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdeqnot.1	⊢ CondEq ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		cdeqim.1	⊢ CondEq ( 𝑥 = 𝑦 → ( 𝜒 ↔ 𝜃 ) )
3	1	cdeqri	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4	2	cdeqri	⊢ ( 𝑥 = 𝑦 → ( 𝜒 ↔ 𝜃 ) )
5	3 4	imbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜃 ) ) )
6	5	cdeqi	⊢ CondEq ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜃 ) ) )