Theorem spimfv

Description: Specialization, using implicit substitution. Version of spim with a disjoint variable condition, which does not require ax-13 . See spimvw for a version with two disjoint variable conditions, requiring fewer axioms, and spimv for another variant. (Contributed by NM, 10-Jan-1993) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	spimfv.nf	$$⊢Ⅎx\psi$$
	spimfv.1	$$⊢x=y\to \left(\phi \to \psi \right)$$
Assertion	spimfv	$$⊢\forall x\phi \to \psi$$

Proof

Step	Hyp	Ref	Expression
1		spimfv.nf	$$⊢Ⅎx\psi$$
2		spimfv.1	$$⊢x=y\to \left(\phi \to \psi \right)$$
3		ax6ev	$$⊢\exists xx=y$$
4	3 2	eximii	$$⊢\exists x\left(\phi \to \psi \right)$$
5	1 4	19.36i	$$⊢\forall x\phi \to \psi$$