Theorem bnj1275

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1275.1	$$⊢\phi \to \exists x\left(\psi \wedge \chi \right)$$
	bnj1275.2	$$⊢\phi \to \forall x\phi$$
Assertion	bnj1275	$$⊢\phi \to \exists x\left(\phi \wedge \psi \wedge \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bnj1275.1	$$⊢\phi \to \exists x\left(\psi \wedge \chi \right)$$
2		bnj1275.2	$$⊢\phi \to \forall x\phi$$
3	2 1	bnj596	$$⊢\phi \to \exists x\left(\phi \wedge \left(\psi \wedge \chi \right)\right)$$
4		3anass	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \left(\phi \wedge \left(\psi \wedge \chi \right)\right)$$
5	3 4	bnj1198	$$⊢\phi \to \exists x\left(\phi \wedge \psi \wedge \chi \right)$$