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Theorem jcab 570
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 107 . . . 4 ((𝜓𝜒) → 𝜓)
21imim2i 12 . . 3 ((𝜑 → (𝜓𝜒)) → (𝜑𝜓))
3 simpr 108 . . . 4 ((𝜓𝜒) → 𝜒)
43imim2i 12 . . 3 ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))
52, 4jca 300 . 2 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) ∧ (𝜑𝜒)))
6 pm3.43 569 . 2 (((𝜑𝜓) ∧ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
75, 6impbii 124 1 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  pm4.76  571  pm5.44  872  2eu4  2041  ssconb  3133  ssin  3222  raaan  3388  tfri3  6132  isprm2  11373
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