Theorem barbarilem 2752

Description: Lemma for barbari 2753 and the other Aristotelian syllogisms with existential assumption. (Contributed by BJ, 16-Sep-2022.)

Hypotheses

Ref	Expression
barbarilem.min	⊢ ∃𝑥𝜑
barbarilem.maj	⊢ ∀𝑥(𝜑 → 𝜓)

Assertion

Ref	Expression
barbarilem	⊢ ∃𝑥(𝜑 ∧ 𝜓)

Proof of Theorem barbarilem

Step	Hyp	Ref	Expression
1		barbarilem.maj	. 2 ⊢ ∀𝑥(𝜑 → 𝜓)
2		barbarilem.min	. 2 ⊢ ∃𝑥𝜑
3		exintr 1892	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
4	1, 2, 3	mp2 9	1 ⊢ ∃𝑥(𝜑 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1534  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780

This theorem is referenced by:  barbari  2753  cesaro  2762  camestros  2763  calemos  2774