Theorem impel

Description: An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019) (Proof shortened by Wolf Lammen, 2-Sep-2020)

	Ref	Expression
Hypotheses	impel.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	impel.2	⊢ ( 𝜃 → 𝜓 )
Assertion	impel	⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		impel.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		impel.2	⊢ ( 𝜃 → 𝜓 )
3	2 1	syl5	⊢ ( 𝜑 → ( 𝜃 → 𝜒 ) )
4	3	imp	⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜒 )