Theorem atbiffatnnb 44407

Description: If a implies b, then a implies not not b. (Contributed by Jarvin Udandy, 28-Aug-2016.)

Assertion

Ref	Expression
atbiffatnnb	⊢ ((𝜑 → 𝜓) → (𝜑 → ¬ ¬ 𝜓))

Proof of Theorem atbiffatnnb

Step	Hyp	Ref	Expression
1		idd 24	. . 3 ⊢ (𝜑 → (𝜓 → 𝜓))
2		notnotb 315	. . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
3	1, 2	syl6ib 250	. 2 ⊢ (𝜑 → (𝜓 → ¬ ¬ 𝜓))
4	3	a2i 14	1 ⊢ ((𝜑 → 𝜓) → (𝜑 → ¬ ¬ 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 depends on definitions:  df-bi 206

This theorem is referenced by:  atbiffatnnbalt  44409