Theorem speivw

Description: Version of spei with a disjoint variable condition, which does not require ax-13 (neither ax-7 nor ax-12 ). (Contributed by BJ, 31-May-2019)

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

Proof

Step	Hyp	Ref	Expression
1		speivw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		speivw.2	$$⊢\psi$$
3	1	biimprd	$$⊢x=y\to \left(\psi \to \phi \right)$$
4	3 2	speiv	$$⊢\exists x\phi$$