Theorem pm4.14 819

Description: Theorem *4.14 of [WhiteheadRussell] p. 117. Related to con34b 319. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)

Assertion

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

Proof of Theorem pm4.14

Step	Hyp	Ref	Expression
1		con34b 319	. . 3 ⊢ ((𝜓 → 𝜒) ↔ (¬ 𝜒 → ¬ 𝜓))
2	1	imbi2i 339	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
3		impexp 456	. 2 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒)))
4		impexp 456	. 2 ⊢ (((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓) ↔ (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
5	2, 3, 4	3bitr4i 306	1 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401

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-an 402

This theorem is used by:  pm3.37  820  ndvdssub  16492