Theorem imimorbdc 891

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		bi2.04 247	. 2 ⊢ (((𝜓 → 𝜒) → (𝜑 → 𝜒)) ↔ (𝜑 → ((𝜓 → 𝜒) → 𝜒)))
2		dfor2dc 890	. . 3 ⊢ (DECID 𝜓 → ((𝜓 ∨ 𝜒) ↔ ((𝜓 → 𝜒) → 𝜒)))
3	2	imbi2d 229	. 2 ⊢ (DECID 𝜓 → ((𝜑 → (𝜓 ∨ 𝜒)) ↔ (𝜑 → ((𝜓 → 𝜒) → 𝜒))))
4	1, 3	bitr4id 198	1 ⊢ (DECID 𝜓 → (((𝜓 → 𝜒) → (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∨ 𝜒))))

Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 703  DECID wdc 829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-dc 830

This theorem is referenced by: (None)