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Theorem 4cases 915
 Description: Inference eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 25-Oct-2003.)
Hypotheses
Ref Expression
4cases.1 ((φ ψ) → χ)
4cases.2 ((φ ¬ ψ) → χ)
4cases.3 ((¬ φ ψ) → χ)
4cases.4 ((¬ φ ¬ ψ) → χ)
Assertion
Ref Expression
4cases χ

Proof of Theorem 4cases
StepHypRef Expression
1 4cases.1 . . 3 ((φ ψ) → χ)
2 4cases.3 . . 3 ((¬ φ ψ) → χ)
31, 2pm2.61ian 765 . 2 (ψχ)
4 4cases.2 . . 3 ((φ ¬ ψ) → χ)
5 4cases.4 . . 3 ((¬ φ ¬ ψ) → χ)
64, 5pm2.61ian 765 . 2 ψχ)
73, 6pm2.61i 156 1 χ
 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:  4casesdan  916  ax11eq  2193  ax11el  2194
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