Theorem axie2

Description: A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017)

	Ref	Expression
Assertion	axie2	$$⊢\forall x\left(\psi \to \forall x\psi \right)\to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nf5	$$⊢Ⅎx\psi \leftrightarrow \forall x\left(\psi \to \forall x\psi \right)$$
2		19.23t	$$⊢Ⅎx\psi \to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \psi \right)\right)$$
3	1 2	sylbir	$$⊢\forall x\left(\psi \to \forall x\psi \right)\to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \psi \right)\right)$$