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Theorem 4cases 989
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 ((𝜑𝜓) → 𝜒)
4cases.2 ((𝜑 ∧ ¬ 𝜓) → 𝜒)
4cases.3 ((¬ 𝜑𝜓) → 𝜒)
4cases.4 ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)
Assertion
Ref Expression
4cases 𝜒

Proof of Theorem 4cases
StepHypRef Expression
1 4cases.1 . . 3 ((𝜑𝜓) → 𝜒)
2 4cases.3 . . 3 ((¬ 𝜑𝜓) → 𝜒)
31, 2pm2.61ian 830 . 2 (𝜓𝜒)
4 4cases.2 . . 3 ((𝜑 ∧ ¬ 𝜓) → 𝜒)
5 4cases.4 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)
64, 5pm2.61ian 830 . 2 𝜓𝜒)
73, 6pm2.61i 176 1 𝜒
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384
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 197  df-an 386
This theorem is referenced by:  4casesdan  990  suc11reg  8460  hasheqf1oi  13080  hasheqf1oiOLD  13081  fvprmselgcd1  15673  axlowdimlem15  25736  ax12eq  33706  ax12el  33707  cdleme27a  35135
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