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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