Theorem jcab 522

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 102	. . . 4 ⊢ ((ψ ∧ χ) → ψ)
2	1	imim2i 12	. . 3 ⊢ ((φ → (ψ ∧ χ)) → (φ → ψ))
3		simpr 103	. . . 4 ⊢ ((ψ ∧ χ) → χ)
4	3	imim2i 12	. . 3 ⊢ ((φ → (ψ ∧ χ)) → (φ → χ))
5	2, 4	jca 290	. 2 ⊢ ((φ → (ψ ∧ χ)) → ((φ → ψ) ∧ (φ → χ)))
6		pm3.43 521	. 2 ⊢ (((φ → ψ) ∧ (φ → χ)) → (φ → (ψ ∧ χ)))
7	5, 6	impbii 117	1 ⊢ ((φ → (ψ ∧ χ)) ↔ ((φ → ψ) ∧ (φ → χ)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  pm3.43OLD  523  pm4.76  524  pm5.44  822  2eu4  1975  ssconb  3053  ssin  3136  raaan  3306  tfri3  5875