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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 → (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Proof of Theorem jcab
StepHypRef Expression
1 simpl 483 . . . 4 ((𝜓𝜒) → 𝜓)
21imim2i 16 . . 3 ((𝜑 → (𝜓𝜒)) → (𝜑𝜓))
3 simpr 485 . . . 4 ((𝜓𝜒) → 𝜒)
43imim2i 16 . . 3 ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))
52, 4jca 512 . 2 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) ∧ (𝜑𝜒)))
6 pm3.43 474 . 2 (((𝜑𝜓) ∧ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
75, 6impbii 208 1 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
Colors of variables: wff setvar class
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  1004  2mo2  2643  ssconb  4137  ssin  4230  2reu4lem  4525  tfr3  8398  trclfvcotr  14955  isprm2  16618  lgsquad2lem2  26885  ostthlem2  27128  pclclN  38757  ifpbibib  42251  elmapintrab  42317  elinintrab  42318  ismnuprim  43043
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