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	⊢ ( ∃ 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		ax5e	⊢ ( ∃ 𝑥 𝜓 → 𝜓 )
4	2 3	syl6	⊢ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → 𝜓 ) )
5	1 4	impbid2	⊢ ( ∃ 𝑥 𝜑 → ( 𝜓 ↔ ∀ 𝑥 𝜓 ) )