Theorem bnj1023

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

	Ref	Expression
Hypotheses	bnj1023.1	$$⊢\exists x\left(\phi \to \psi \right)$$
	bnj1023.2	$$⊢\psi \to \chi$$
Assertion	bnj1023	$$⊢\exists x\left(\phi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bnj1023.1	$$⊢\exists x\left(\phi \to \psi \right)$$
2		bnj1023.2	$$⊢\psi \to \chi$$
3	2	a1i	$$⊢\left(\phi \to \psi \right)\to \left(\psi \to \chi \right)$$
4	3	ax-gen	$$⊢\forall x\left(\left(\phi \to \psi \right)\to \left(\psi \to \chi \right)\right)$$
5		exintr	$$⊢\forall x\left(\left(\phi \to \psi \right)\to \left(\psi \to \chi \right)\right)\to \left(\exists x\left(\phi \to \psi \right)\to \exists x\left(\left(\phi \to \psi \right)\wedge \left(\psi \to \chi \right)\right)\right)$$
6	4 1 5	mp2	$$⊢\exists x\left(\left(\phi \to \psi \right)\wedge \left(\psi \to \chi \right)\right)$$
7		pm3.33	$$⊢\left(\left(\phi \to \psi \right)\wedge \left(\psi \to \chi \right)\right)\to \left(\phi \to \chi \right)$$
8	6 7	bnj101	$$⊢\exists x\left(\phi \to \chi \right)$$