This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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	$⊢ \forall x (φ \to ψ) \to (\exists x φ \to \exists x (φ \land ψ))$

Proof

Step	Hyp	Ref	Expression
1		ancl	$⊢ (φ \to ψ) \to (φ \to (φ \land ψ))$
2	1	aleximi	$⊢ \forall x (φ \to ψ) \to (\exists x φ \to \exists x (φ \land ψ))$