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	⊢ ( ∃ 𝑥 𝜑 → ( 𝜓 ↔ ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-5	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
2		bj-axdd2	⊢ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∃ 𝑥 𝜓 ) )
3	1 2	syl5	⊢ ( ∃ 𝑥 𝜑 → ( 𝜓 → ∃ 𝑥 𝜓 ) )
4		ax5e	⊢ ( ∃ 𝑥 𝜓 → 𝜓 )
5	3 4	impbid1	⊢ ( ∃ 𝑥 𝜑 → ( 𝜓 ↔ ∃ 𝑥 𝜓 ) )