Theorem equs5av

Description: A property related to substitution that replaces the distinctor from equs5 to a disjoint variable condition. Version of equs5a with a disjoint variable condition, which does not require ax-13 . See also sbalex . (Contributed by NM, 2-Feb-2007) (Revised by Gino Giotto, 15-Dec-2023)

	Ref	Expression
Assertion	equs5av	$$⊢\exists x\left(x=y\wedge \forall y\phi \right)\to \forall x\left(x=y\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfa1	$$⊢Ⅎx\forall x\left(x=y\to \phi \right)$$
2		ax12v2	$$⊢x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)$$
3	2	spsd	$$⊢x=y\to \left(\forall y\phi \to \forall x\left(x=y\to \phi \right)\right)$$
4	3	imp	$$⊢\left(x=y\wedge \forall y\phi \right)\to \forall x\left(x=y\to \phi \right)$$
5	1 4	exlimi	$$⊢\exists x\left(x=y\wedge \forall y\phi \right)\to \forall x\left(x=y\to \phi \right)$$