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Theorem jcab 521
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 486 . . . 4 ((𝜓𝜒) → 𝜓)
21imim2i 16 . . 3 ((𝜑 → (𝜓𝜒)) → (𝜑𝜓))
3 simpr 488 . . . 4 ((𝜓𝜒) → 𝜒)
43imim2i 16 . . 3 ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))
52, 4jca 515 . 2 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) ∧ (𝜑𝜒)))
6 pm3.43 477 . 2 (((𝜑𝜓) ∧ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
75, 6impbii 212 1 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399
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 210  df-an 400
This theorem is referenced by:  pm4.76  522  pm5.44  546  ordi  1006  2mo2  2648  ssconb  4057  ssin  4150  2reu4lem  4442  tfr3  8140  trclfvcotr  14577  isprm2  16244  lgsquad2lem2  26271  ostthlem2  26514  pclclN  37647  ifpbibib  40810  elmapintrab  40868  elinintrab  40869  ismnuprim  41593
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