Theorem exintr

Description: Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993) (Proof shortened by BJ, 16-Sep-2022)

	Ref	Expression
Assertion	exintr	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ancl	⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) )
2	1	aleximi	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )