Theorem ainaiaandna 47702

Description: Given a, a implies it is not the case a implies a self contradiction. (Contributed by Jarvin Udandy, 7-Sep-2020.)

Hypothesis

Ref	Expression
ainaiaandna.1	⊢ 𝜑

Assertion

Ref	Expression
ainaiaandna	⊢ (𝜑 → ¬ (𝜑 → (𝜑 ∧ ¬ 𝜑)))

Proof of Theorem ainaiaandna

Step	Hyp	Ref	Expression
1		ainaiaandna.1	. . 3 ⊢ 𝜑
2	1	atnaiana 47701	. 2 ⊢ ¬ (𝜑 → (𝜑 ∧ ¬ 𝜑))
3	2	a1i 11	1 ⊢ (𝜑 → ¬ (𝜑 → (𝜑 ∧ ¬ 𝜑)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583

This theorem is used by: (None)