Theorem cdeqim 3739

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

Hypotheses

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

Assertion

Ref	Expression
cdeqim	⊢ CondEq(𝑥 = 𝑦 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))

Proof of Theorem cdeqim

Step	Hyp	Ref	Expression
1		cdeqnot.1	. . . 4 ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	cdeqri 3732	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3		cdeqim.1	. . . 4 ⊢ CondEq(𝑥 = 𝑦 → (𝜒 ↔ 𝜃))
4	3	cdeqri 3732	. . 3 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜃))
5	2, 4	imbi12d 347	. 2 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))
6	5	cdeqi 3731	1 ⊢ CondEq(𝑥 = 𝑦 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  CondEqwcdeq 3729

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-cdeq 3730

This theorem is used by: (None)