Theorem 3impexp 1366

Description: impexp 433 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.)

Assertion

Ref	Expression
3impexp	⊢ (((φ ∧ ψ ∧ χ) → θ) ↔ (φ → (ψ → (χ → θ))))

Proof of Theorem 3impexp

Step	Hyp	Ref	Expression
1		id 19	. . 3 ⊢ (((φ ∧ ψ ∧ χ) → θ) → ((φ ∧ ψ ∧ χ) → θ))
2	1	3expd 1168	. 2 ⊢ (((φ ∧ ψ ∧ χ) → θ) → (φ → (ψ → (χ → θ))))
3		id 19	. . 3 ⊢ ((φ → (ψ → (χ → θ))) → (φ → (ψ → (χ → θ))))
4	3	3impd 1165	. 2 ⊢ ((φ → (ψ → (χ → θ))) → ((φ ∧ ψ ∧ χ) → θ))
5	2, 4	impbii 180	1 ⊢ (((φ ∧ ψ ∧ χ) → θ) ↔ (φ → (ψ → (χ → θ))))

Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934

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  df-3an 936

This theorem is referenced by:  3impexpbicom  1367