Theorem efald 1560

Description: Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)

Hypothesis

Ref	Expression
efald.1	⊢ ((𝜑 ∧ ¬ 𝜓) → ⊥)

Assertion

Ref	Expression
efald	⊢ (𝜑 → 𝜓)

Proof of Theorem efald

Step	Hyp	Ref	Expression
1		efald.1	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → ⊥)
2	1	inegd 1559	. 2 ⊢ (𝜑 → ¬ ¬ 𝜓)
3	2	notnotrd 133	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ⊥wfal 1551

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-an 397  df-tru 1542  df-fal 1552

This theorem is referenced by:  ablsimpgfind  19713  efald2  36236