Theorem efald 1564

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 1563	. 2 ⊢ (𝜑 → ¬ ¬ 𝜓)
3	2	notnotrd 135	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ⊥wfal 1555

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 210  df-an 400  df-tru 1546  df-fal 1556

This theorem is referenced by:  ablsimpgfind  19497  efald2  35973