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Theorem nnexmid 835
Description: Double negation of excluded middle. Intuitionistic logic refutes the negation of excluded middle (but does not prove excluded middle) for any formula. Can also be proved quickly from bj-nnor 13053 as in bj-nndcALT 13070. (Contributed by BJ, 9-Oct-2019.)
Assertion
Ref Expression
nnexmid ¬ ¬ (𝜑 ∨ ¬ 𝜑)

Proof of Theorem nnexmid
StepHypRef Expression
1 pm3.24 682 . 2 ¬ (¬ 𝜑 ∧ ¬ ¬ 𝜑)
2 ioran 741 . 2 (¬ (𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∧ ¬ ¬ 𝜑))
31, 2mtbir 660 1 ¬ ¬ (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wo 697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  nndc  836  exmid1stab  13302
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