Theorem jcab 518

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 483	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜓)
2	1	imim2i 16	. . 3 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → (𝜑 → 𝜓))
3		simpr 485	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜒)
4	3	imim2i 16	. . 3 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → (𝜑 → 𝜒))
5	2, 4	jca 512	. 2 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))
6		pm3.43 474	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ∧ 𝜒)))
7	5, 6	impbii 208	1 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396

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 206  df-an 397

This theorem is referenced by:  pm4.76  519  pm5.44  543  ordi  1003  2mo2  2649  ssconb  4072  ssin  4164  2reu4lem  4456  tfr3  8230  trclfvcotr  14720  isprm2  16387  lgsquad2lem2  26533  ostthlem2  26776  pclclN  37905  ifpbibib  41117  elmapintrab  41184  elinintrab  41185  ismnuprim  41912