Theorem negel 38111

Description: An inference for negation elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)

Hypotheses

Ref	Expression
negel.1	⊢ (𝜓 → 𝜒)
negel.2	⊢ (𝜑 → ¬ 𝜒)

Assertion

Ref	Expression
negel	⊢ ((𝜑 ∧ 𝜓) → ⊥)

Proof of Theorem negel

Step	Hyp	Ref	Expression
1		negel.1	. . 3 ⊢ (𝜓 → 𝜒)
2	1	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3		negel.2	. . 3 ⊢ (𝜑 → ¬ 𝜒)
4	3	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜒)
5	2, 4	pm2.21fal 1561	1 ⊢ ((𝜑 ∧ 𝜓) → ⊥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ⊥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 207  df-an 396

This theorem is referenced by: (None)