Theorem jcab 833

Description: Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 27-Nov-2013.)

Assertion

Ref	Expression
jcab	⊢ ((φ → (ψ ∧ χ)) ↔ ((φ → ψ) ∧ (φ → χ)))

Proof of Theorem jcab

Step	Hyp	Ref	Expression
1		simpl 443	. . . 4 ⊢ ((ψ ∧ χ) → ψ)
2	1	imim2i 13	. . 3 ⊢ ((φ → (ψ ∧ χ)) → (φ → ψ))
3		simpr 447	. . . 4 ⊢ ((ψ ∧ χ) → χ)
4	3	imim2i 13	. . 3 ⊢ ((φ → (ψ ∧ χ)) → (φ → χ))
5	2, 4	jca 518	. 2 ⊢ ((φ → (ψ ∧ χ)) → ((φ → ψ) ∧ (φ → χ)))
6		pm3.43 832	. 2 ⊢ (((φ → ψ) ∧ (φ → χ)) → (φ → (ψ ∧ χ)))
7	5, 6	impbii 180	1 ⊢ ((φ → (ψ ∧ χ)) ↔ ((φ → 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:  ordi  834  pm4.76  836  pm5.44  877  2eu4  2287  ssconb  3400  ssin  3478