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Theorem atnaiana 41607
Description: Given a, it is not the case a implies a self contradiction. (Contributed by Jarvin Udandy, 7-Sep-2020.)
Hypothesis
Ref Expression
atnaiana.1 𝜑
Assertion
Ref Expression
atnaiana ¬ (𝜑 → (𝜑 ∧ ¬ 𝜑))

Proof of Theorem atnaiana
StepHypRef Expression
1 atnaiana.1 . . . 4 𝜑
21bitru 1644 . . 3 (𝜑 ↔ ⊤)
3 pm3.24 389 . . . 4 ¬ (𝜑 ∧ ¬ 𝜑)
43bifal 1645 . . 3 ((𝜑 ∧ ¬ 𝜑) ↔ ⊥)
52, 4aifftbifffaibif 41605 . 2 ((𝜑 → (𝜑 ∧ ¬ 𝜑)) ↔ ⊥)
65aisfina 41582 1 ¬ (𝜑 → (𝜑 ∧ ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382
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 383  df-tru 1634  df-fal 1637
This theorem is referenced by:  ainaiaandna  41608  confun5  41627
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