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Theorem 3ornot23 37532
Description: If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 37900. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
3ornot23 ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒𝜑𝜓) → 𝜒))

Proof of Theorem 3ornot23
StepHypRef Expression
1 idd 24 . . 3 𝜑 → (𝜒𝜒))
2 pm2.21 118 . . 3 𝜑 → (𝜑𝜒))
3 pm2.21 118 . . 3 𝜓 → (𝜓𝜒))
41, 2, 33jaao 1387 . 2 ((¬ 𝜑 ∧ ¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒𝜑𝜓) → 𝜒))
543anidm12 1374 1 ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒𝜑𝜓) → 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3o 1029
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 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032
This theorem is referenced by:  tratrb  37563  tratrbVD  37915
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