Theorem imimorbdc 828

Description: Simplify an implication between implications, for a decidable proposition. (Contributed by Jim Kingdon, 18-Mar-2018.)

Assertion

Ref	Expression
imimorbdc	⊢ (DECID 𝜓 → (((𝜓 → 𝜒) → (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∨ 𝜒))))

Proof of Theorem imimorbdc

Step	Hyp	Ref	Expression
1		dfor2dc 827	. . 3 ⊢ (DECID 𝜓 → ((𝜓 ∨ 𝜒) ↔ ((𝜓 → 𝜒) → 𝜒)))
2	1	imbi2d 228	. 2 ⊢ (DECID 𝜓 → ((𝜑 → (𝜓 ∨ 𝜒)) ↔ (𝜑 → ((𝜓 → 𝜒) → 𝜒))))
3		bi2.04 246	. 2 ⊢ (((𝜓 → 𝜒) → (𝜑 → 𝜒)) ↔ (𝜑 → ((𝜓 → 𝜒) → 𝜒)))
4	2, 3	syl6rbbr 197	1 ⊢ (DECID 𝜓 → (((𝜓 → 𝜒) → (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∨ 𝜒))))

Syntax hints:   → wi 4   ↔ wb 103   ∨ wo 661  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by: (None)