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Theorem jcab 833
 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)) ↔ ((φψ) (φχ)))

Proof of Theorem jcab
StepHypRef Expression
1 simpl 443 . . . 4 ((ψ χ) → ψ)
21imim2i 13 . . 3 ((φ → (ψ χ)) → (φψ))
3 simpr 447 . . . 4 ((ψ χ) → χ)
43imim2i 13 . . 3 ((φ → (ψ χ)) → (φχ))
52, 4jca 518 . 2 ((φ → (ψ χ)) → ((φψ) (φχ)))
6 pm3.43 832 . 2 (((φψ) (φχ)) → (φ → (ψ χ)))
75, 6impbii 180 1 ((φ → (ψ χ)) ↔ ((φψ) (φχ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358 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 177  df-an 360 This theorem is referenced by:  ordi  834  pm4.76  836  pm5.44  877  2eu4  2287  ssconb  3399  ssin  3477
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