Theorem jcab 593

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 108	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜓)
2	1	imim2i 12	. . 3 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → (𝜑 → 𝜓))
3		simpr 109	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜒)
4	3	imim2i 12	. . 3 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → (𝜑 → 𝜒))
5	2, 4	jca 304	. 2 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))
6		pm3.43 592	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ∧ 𝜒)))
7	5, 6	impbii 125	1 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm4.76  594  pm5.44  915  2eu4  2107  ssconb  3255  ssin  3344  raaan  3515  tfri3  6335  omniwomnimkv  7131  isprm2  12049