Theorem pm4.87 839

Description: Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.)

Assertion

Ref	Expression
pm4.87	⊢ (((((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒))) ∧ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜓 → (𝜑 → 𝜒)))) ∧ ((𝜓 → (𝜑 → 𝜒)) ↔ ((𝜓 ∧ 𝜑) → 𝜒)))

Proof of Theorem pm4.87

Step	Hyp	Ref	Expression
1		impexp 450	. . 3 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒)))
2		bi2.04 388	. . 3 ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜓 → (𝜑 → 𝜒)))
3	1, 2	pm3.2i 470	. 2 ⊢ ((((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒))) ∧ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜓 → (𝜑 → 𝜒))))
4		impexp 450	. . 3 ⊢ (((𝜓 ∧ 𝜑) → 𝜒) ↔ (𝜓 → (𝜑 → 𝜒)))
5	4	bicomi 223	. 2 ⊢ ((𝜓 → (𝜑 → 𝜒)) ↔ ((𝜓 ∧ 𝜑) → 𝜒))
6	3, 5	pm3.2i 470	1 ⊢ (((((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒))) ∧ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜓 → (𝜑 → 𝜒)))) ∧ ((𝜓 → (𝜑 → 𝜒)) ↔ ((𝜓 ∧ 𝜑) → 𝜒)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 396

This theorem is referenced by: (None)