Theorem pm10.56

Description: Theorem *10.56 in WhiteheadRussell p. 156. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm10.56	$$⊢\left(\forall x\left(\phi \to \psi \right)\wedge \exists x\left(\phi \wedge \chi \right)\right)\to \exists x\left(\psi \wedge \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		pm3.45	$$⊢\left(\phi \to \psi \right)\to \left(\left(\phi \wedge \chi \right)\to \left(\psi \wedge \chi \right)\right)$$
2	1	aleximi	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\exists x\left(\phi \wedge \chi \right)\to \exists x\left(\psi \wedge \chi \right)\right)$$
3	2	imp	$$⊢\left(\forall x\left(\phi \to \psi \right)\wedge \exists x\left(\phi \wedge \chi \right)\right)\to \exists x\left(\psi \wedge \chi \right)$$