Theorem pm4.87 567

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 433	. . 3 ⊢ (((φ ∧ ψ) → χ) ↔ (φ → (ψ → χ)))
2		bi2.04 350	. . 3 ⊢ ((φ → (ψ → χ)) ↔ (ψ → (φ → χ)))
3	1, 2	pm3.2i 441	. 2 ⊢ ((((φ ∧ ψ) → χ) ↔ (φ → (ψ → χ))) ∧ ((φ → (ψ → χ)) ↔ (ψ → (φ → χ))))
4		impexp 433	. . 3 ⊢ (((ψ ∧ φ) → χ) ↔ (ψ → (φ → χ)))
5	4	bicomi 193	. 2 ⊢ ((ψ → (φ → χ)) ↔ ((ψ ∧ φ) → χ))
6	3, 5	pm3.2i 441	1 ⊢ (((((φ ∧ ψ) → χ) ↔ (φ → (ψ → χ))) ∧ ((φ → (ψ → χ)) ↔ (ψ → (φ → χ)))) ∧ ((ψ → (φ → χ)) ↔ ((ψ ∧ φ) → χ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358

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 177  df-an 360

This theorem is referenced by: (None)