Theorem exintrbi

Description: Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006)

	Ref	Expression
Assertion	exintrbi	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \leftrightarrow \exists x\left(\phi \wedge \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		abai	$$⊢\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \left(\phi \to \psi \right)\right)$$
2	1	rbaibr	$$⊢\left(\phi \to \psi \right)\to \left(\phi \leftrightarrow \left(\phi \wedge \psi \right)\right)$$
3	2	alexbii	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \leftrightarrow \exists x\left(\phi \wedge \psi \right)\right)$$