Theorem bnj1101

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

	Ref	Expression
Hypotheses	bnj1101.1	$$⊢\exists x\left(\phi \to \psi \right)$$
	bnj1101.2	$$⊢\chi \to \phi$$
Assertion	bnj1101	$$⊢\exists x\left(\chi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bnj1101.1	$$⊢\exists x\left(\phi \to \psi \right)$$
2		bnj1101.2	$$⊢\chi \to \phi$$
3		pm3.42	$$⊢\left(\phi \to \psi \right)\to \left(\left(\chi \wedge \phi \right)\to \psi \right)$$
4	1 3	bnj101	$$⊢\exists x\left(\left(\chi \wedge \phi \right)\to \psi \right)$$
5	2	pm4.71i	$$⊢\chi \leftrightarrow \left(\chi \wedge \phi \right)$$
6	5	imbi1i	$$⊢\left(\chi \to \psi \right)\leftrightarrow \left(\left(\chi \wedge \phi \right)\to \psi \right)$$
7	6	exbii	$$⊢\exists x\left(\chi \to \psi \right)\leftrightarrow \exists x\left(\left(\chi \wedge \phi \right)\to \psi \right)$$
8	4 7	mpbir	$$⊢\exists x\left(\chi \to \psi \right)$$