Theorem pm4.14dc 787

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 765	. . 3 ⊢ (DECID 𝜒 → ((𝜓 → 𝜒) ↔ (¬ 𝜒 → ¬ 𝜓)))
2	1	imbi2d 219	. 2 ⊢ (DECID 𝜒 → ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜑 → (¬ 𝜒 → ¬ 𝜓))))
3		impexp 250	. 2 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒)))
4		impexp 250	. 2 ⊢ (((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓) ↔ (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
5	2, 3, 4	3bitr4g 212	1 ⊢ (DECID 𝜒 → (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  pm3.37dc  788