Theorem negel

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

	Ref	Expression
Hypotheses	negel.1	⊢ ( 𝜓 → 𝜒 )
	negel.2	⊢ ( 𝜑 → ¬ 𝜒 )
Assertion	negel	⊢ ( ( 𝜑 ∧ 𝜓 ) → ⊥ )

Proof

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