Theorem pm4.14dc 875

Description: Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.)

Assertion

Ref	Expression
pm4.14dc	⊢ (DECID 𝜒 → (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)))

Proof of Theorem pm4.14dc

Step	Hyp	Ref	Expression
1		con34bdc 856	. . 3 ⊢ (DECID 𝜒 → ((𝜓 → 𝜒) ↔ (¬ 𝜒 → ¬ 𝜓)))
2	1	imbi2d 229	. 2 ⊢ (DECID 𝜒 → ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜑 → (¬ 𝜒 → ¬ 𝜓))))
3		impexp 261	. 2 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒)))
4		impexp 261	. 2 ⊢ (((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓) ↔ (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
5	2, 3, 4	3bitr4g 222	1 ⊢ (DECID 𝜒 → (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 819

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 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820

This theorem is referenced by: (None)