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Metamath Proof Explorer

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 )