Theorem bj-equs45fv

Description: Version of equs45f with a disjoint variable condition, which does not require ax-13 . Note that the version of equs5 with a disjoint variable condition is actually sbalex (up to adding a superfluous antecedent). (Contributed by BJ, 11-Sep-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	bj-equs45fv.1	$$⊢Ⅎy\phi$$
Assertion	bj-equs45fv	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \forall x\left(x=y\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-equs45fv.1	$$⊢Ⅎy\phi$$
2	1	nf5ri	$$⊢\phi \to \forall y\phi$$
3	2	anim2i	$$⊢\left(x=y\wedge \phi \right)\to \left(x=y\wedge \forall y\phi \right)$$
4	3	eximi	$$⊢\exists x\left(x=y\wedge \phi \right)\to \exists x\left(x=y\wedge \forall y\phi \right)$$
5		equs5av	$$⊢\exists x\left(x=y\wedge \forall y\phi \right)\to \forall x\left(x=y\to \phi \right)$$
6	4 5	syl	$$⊢\exists x\left(x=y\wedge \phi \right)\to \forall x\left(x=y\to \phi \right)$$
7		equs4v	$$⊢\forall x\left(x=y\to \phi \right)\to \exists x\left(x=y\wedge \phi \right)$$
8	6 7	impbii	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \forall x\left(x=y\to \phi \right)$$