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Theorem jcab 575
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 108 . . . 4 ((𝜓𝜒) → 𝜓)
21imim2i 12 . . 3 ((𝜑 → (𝜓𝜒)) → (𝜑𝜓))
3 simpr 109 . . . 4 ((𝜓𝜒) → 𝜒)
43imim2i 12 . . 3 ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))
52, 4jca 302 . 2 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) ∧ (𝜑𝜒)))
6 pm3.43 574 . 2 (((𝜑𝜓) ∧ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
75, 6impbii 125 1 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  pm4.76  576  pm5.44  893  2eu4  2068  ssconb  3177  ssin  3266  raaan  3437  tfri3  6230  isprm2  11705
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