Theorem spimed

Description: Deduction version of spime . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker spimedv if possible. (Contributed by NM, 14-May-1993) (Revised by Mario Carneiro, 3-Oct-2016) (Proof shortened by Wolf Lammen, 19-Feb-2018) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		spimed.1	$$⊢\chi \to Ⅎx\phi$$
2		spimed.2	$$⊢x=y\to \left(\phi \to \psi \right)$$
3	1	nf5rd	$$⊢\chi \to \left(\phi \to \forall x\phi \right)$$
4		ax6e	$$⊢\exists xx=y$$
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	$$⊢\chi \to \left(\phi \to \exists x\psi \right)$$