Theorem imsym1 34586

Description: A symmetry with →.

See negsym1 34585 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

Assertion

Ref	Expression
imsym1	⊢ ((𝜓 → (𝜓 → ⊥)) → (𝜓 → 𝜑))

Proof of Theorem imsym1

Step	Hyp	Ref	Expression
1		pm2.21 123	. 2 ⊢ (¬ 𝜓 → (𝜓 → 𝜑))
2		falim 1558	. . 3 ⊢ (⊥ → 𝜑)
3	2	imim2i 16	. 2 ⊢ ((𝜓 → ⊥) → (𝜓 → 𝜑))
4	1, 3	ja 186	1 ⊢ ((𝜓 → (𝜓 → ⊥)) → (𝜓 → 𝜑))

Syntax hints:   → wi 4  ⊥wfal 1553

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 206  df-tru 1544  df-fal 1554

This theorem is referenced by: (None)