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Theorem jcab 522
 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 102 . . . 4 ((ψ χ) → ψ)
21imim2i 12 . . 3 ((φ → (ψ χ)) → (φψ))
3 simpr 103 . . . 4 ((ψ χ) → χ)
43imim2i 12 . . 3 ((φ → (ψ χ)) → (φχ))
52, 4jca 290 . 2 ((φ → (ψ χ)) → ((φψ) (φχ)))
6 pm3.43 521 . 2 (((φψ) (φχ)) → (φ → (ψ χ)))
75, 6impbii 117 1 ((φ → (ψ χ)) ↔ ((φψ) (φχ)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  pm3.43OLD  523  pm4.76  524  pm5.44  822  2eu4  1975  ssconb  3053  ssin  3136  raaan  3306  tfri3  5875
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