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Theorem ecase23d 1585
Description: Deduction for elimination by cases. (Contributed by NM, 22-Apr-1994.)
Hypotheses
Ref Expression
ecase23d.1 (𝜑 → ¬ 𝜒)
ecase23d.2 (𝜑 → ¬ 𝜃)
ecase23d.3 (𝜑 → (𝜓𝜒𝜃))
Assertion
Ref Expression
ecase23d (𝜑𝜓)

Proof of Theorem ecase23d
StepHypRef Expression
1 ecase23d.1 . . 3 (𝜑 → ¬ 𝜒)
2 ecase23d.2 . . 3 (𝜑 → ¬ 𝜃)
3 ioran 512 . . 3 (¬ (𝜒𝜃) ↔ (¬ 𝜒 ∧ ¬ 𝜃))
41, 2, 3sylanbrc 701 . 2 (𝜑 → ¬ (𝜒𝜃))
5 ecase23d.3 . . . 4 (𝜑 → (𝜓𝜒𝜃))
6 3orass 1075 . . . 4 ((𝜓𝜒𝜃) ↔ (𝜓 ∨ (𝜒𝜃)))
75, 6sylib 208 . . 3 (𝜑 → (𝜓 ∨ (𝜒𝜃)))
87ord 391 . 2 (𝜑 → (¬ 𝜓 → (𝜒𝜃)))
94, 8mt3d 140 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  w3o 1071
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-or 384  df-an 385  df-3or 1073
This theorem is referenced by:  tz7.7  5910  wfrlem10  7593  archiabllem2b  30059  nolt02o  32151  noresle  32152  nosupbnd1lem6  32165  nosupbnd2lem1  32167
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