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Theorem exmid2 32865
Description: An excluded middle law. (Contributed by Giovanni Mascellani, 23-May-2019.)
Hypotheses
Ref Expression
exmid2.1 ((𝜓𝜑) → 𝜒)
exmid2.2 ((¬ 𝜓𝜂) → 𝜒)
Assertion
Ref Expression
exmid2 ((𝜑𝜂) → 𝜒)

Proof of Theorem exmid2
StepHypRef Expression
1 simpl 472 . . . . 5 ((𝜑𝜂) → 𝜑)
21anim2i 591 . . . 4 ((𝜓 ∧ (𝜑𝜂)) → (𝜓𝜑))
32ancoms 468 . . 3 (((𝜑𝜂) ∧ 𝜓) → (𝜓𝜑))
4 exmid2.1 . . 3 ((𝜓𝜑) → 𝜒)
53, 4syl 17 . 2 (((𝜑𝜂) ∧ 𝜓) → 𝜒)
6 simpr 476 . . . . 5 ((𝜑𝜂) → 𝜂)
76anim2i 591 . . . 4 ((¬ 𝜓 ∧ (𝜑𝜂)) → (¬ 𝜓𝜂))
87ancoms 468 . . 3 (((𝜑𝜂) ∧ ¬ 𝜓) → (¬ 𝜓𝜂))
9 exmid2.2 . . 3 ((¬ 𝜓𝜂) → 𝜒)
108, 9syl 17 . 2 (((𝜑𝜂) ∧ ¬ 𝜓) → 𝜒)
115, 10pm2.61dan 828 1 ((𝜑𝜂) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383
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 196  df-an 385
This theorem is referenced by: (None)
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