Theorem inegd 1589

Description: Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)

Hypothesis

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

Assertion

Ref	Expression
inegd	⊢ (𝜑 → ¬ 𝜓)

Proof of Theorem inegd

Step	Hyp	Ref	Expression
1		inegd.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ⊥)
2	1	ex 418	. 2 ⊢ (𝜑 → (𝜓 → ⊥))
3		dfnot 1588	. 2 ⊢ (¬ 𝜓 ↔ (𝜓 → ⊥))
4	2, 3	sylibr 237	1 ⊢ (𝜑 → ¬ 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ⊥wfal 1581

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582

This theorem is used by:  efald  1590  tglndim0  29109  archiabllem2c  33809  rprmirred  34116  ply1dg3rt0irred  34169  lindsun  34310