Theorem barbarilem 2667

Description: Lemma for barbari 2668 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 1894	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
4	1, 2, 3	mp2 9	1 ⊢ ∃𝑥(𝜑 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1538  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1781

This theorem is referenced by:  barbari  2668  cesaro  2677  camestros  2678  calemos  2689