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Theorem 4casesdan 916
 Description: Deduction eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 19-Mar-2013.)
Hypotheses
Ref Expression
4casesdan.1 ((φ (ψ χ)) → θ)
4casesdan.2 ((φ (ψ ¬ χ)) → θ)
4casesdan.3 ((φ ψ χ)) → θ)
4casesdan.4 ((φ ψ ¬ χ)) → θ)
Assertion
Ref Expression
4casesdan (φθ)

Proof of Theorem 4casesdan
StepHypRef Expression
1 4casesdan.1 . . 3 ((φ (ψ χ)) → θ)
21expcom 424 . 2 ((ψ χ) → (φθ))
3 4casesdan.2 . . 3 ((φ (ψ ¬ χ)) → θ)
43expcom 424 . 2 ((ψ ¬ χ) → (φθ))
5 4casesdan.3 . . 3 ((φ ψ χ)) → θ)
65expcom 424 . 2 ((¬ ψ χ) → (φθ))
7 4casesdan.4 . . 3 ((φ ψ ¬ χ)) → θ)
87expcom 424 . 2 ((¬ ψ ¬ χ) → (φθ))
92, 4, 6, 84cases 915 1 (φθ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ 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: (None)
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