Theorem jcab 517

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 482	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜓)
2	1	imim2i 16	. . 3 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → (𝜑 → 𝜓))
3		simpr 484	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜒)
4	3	imim2i 16	. . 3 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → (𝜑 → 𝜒))
5	2, 4	jca 511	. 2 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))
6		pm3.43 473	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ∧ 𝜒)))
7	5, 6	impbii 209	1 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395

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 207  df-an 396

This theorem is referenced by:  pm4.76  518  pm5.44  542  ordi  1007  2mo2  2644  ssconb  4091  ssin  4188  2reu4lem  4471  tfr3  8324  trclfvcotr  14918  isprm2  16595  lgsquad2lem2  27324  ostthlem2  27567  pclclN  40010  ifpbibib  43627  elmapintrab  43693  elinintrab  43694  ismnuprim  44411