Theorem spimedv

Description: Deduction version of spimev . Version of spimed with a disjoint variable condition, which does not require ax-13 . See spime for a non-deduction version. (Contributed by NM, 14-May-1993) (Revised by BJ, 31-May-2019)

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

Proof

Step	Hyp	Ref	Expression
1		spimedv.1	$$⊢\chi \to Ⅎx\phi$$
2		spimedv.2	$$⊢x=y\to \left(\phi \to \psi \right)$$
3	1	nf5rd	$$⊢\chi \to \left(\phi \to \forall x\phi \right)$$
4		ax6ev	$$⊢\exists xx=y$$
5	4 2	eximii	$$⊢\exists x\left(\phi \to \psi \right)$$
6	5	19.35i	$$⊢\forall x\phi \to \exists x\psi$$
7	3 6	syl6	$$⊢\chi \to \left(\phi \to \exists x\psi \right)$$