Theorem barbarilem

Description: Lemma for barbari and the other Aristotelian syllogisms with existential assumption. (Contributed by BJ, 16-Sep-2022)

	Ref	Expression
Hypotheses	barbarilem.min	⊢ ∃ 𝑥 𝜑
	barbarilem.maj	⊢ ∀ 𝑥 ( 𝜑 → 𝜓 )
Assertion	barbarilem	⊢ ∃ 𝑥 ( 𝜑 ∧ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		barbarilem.min	⊢ ∃ 𝑥 𝜑
2		barbarilem.maj	⊢ ∀ 𝑥 ( 𝜑 → 𝜓 )
3		exintr	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )
4	2 1 3	mp2	⊢ ∃ 𝑥 ( 𝜑 ∧ 𝜓 )