Theorem bj-spvew

Description: Version of 19.8v and 19.9v proved from ax-1 -- ax-5 . The antecedent can for instance be proved with the existence axiom extru . (Contributed by BJ, 8-Mar-2026) This could also be proved from bj-spvw using duality, but that proof would not be intuitionistic, contrary to the present one. (Proof modification is discouraged.)

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

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	1 2	syl5	$$⊢\exists x\phi \to \left(\psi \to \exists x\psi \right)$$
4		ax5e	$$⊢\exists x\psi \to \psi$$
5	3 4	impbid1	$$⊢\exists x\phi \to \left(\psi \leftrightarrow \exists x\psi \right)$$