Theorem bj-bijust00 33905

Description: A self-implication does not imply the negation of a self-implication. Most general theorem of which bijust 207 is an instance (bijust0 206 and bj-bijust0ALT 33904 are therefore also instances of it). (Contributed by BJ, 7-Sep-2022.)

Assertion

Ref	Expression
bj-bijust00	⊢ ¬ ((𝜑 → 𝜑) → ¬ (𝜓 → 𝜓))

Proof of Theorem bj-bijust00

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝜑 → 𝜑)
2		id 22	. 2 ⊢ (𝜓 → 𝜓)
3		pm3.2im 162	. 2 ⊢ ((𝜑 → 𝜑) → ((𝜓 → 𝜓) → ¬ ((𝜑 → 𝜑) → ¬ (𝜓 → 𝜓))))
4	1, 2, 3	mp2 9	1 ⊢ ¬ ((𝜑 → 𝜑) → ¬ (𝜓 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by: (None)