Theorem bj-spvw

Description: Version of spvw and 19.3v proved from ax-1 -- ax-5 . The antecedent can for instance be proved with the existence axiom extru . (Contributed by BJ, 8-Mar-2026) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-spvw	$$⊢\exists x\phi \to \left(\psi \leftrightarrow \forall x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-5	$$⊢\psi \to \forall x\psi$$
2		bj-axdd2	$$⊢\exists x\phi \to \left(\forall x\psi \to \exists x\psi \right)$$
3		ax5e	$$⊢\exists x\psi \to \psi$$
4	2 3	syl6	$$⊢\exists x\phi \to \left(\forall x\psi \to \psi \right)$$
5	1 4	impbid2	$$⊢\exists x\phi \to \left(\psi \leftrightarrow \forall x\psi \right)$$