Theorem bj-spvew 36977

Description: Version of 19.8v 1990 and 19.9v 1991 proved from ax-1 6-- ax-5 1917. The antecedent can for instance be proved with the existence axiom extru 1982. (Contributed by BJ, 8-Mar-2026.) This could also be proved from bj-spvw 36976 using duality, but that proof would not be intuitionistic, contrary to the present one. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-spvew	⊢ (∃𝑥𝜑 → (𝜓 ↔ ∃𝑥𝜓))

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-spvew

Step	Hyp	Ref	Expression
1		ax-5 1917	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
2		bj-axdd2 36904	. . 3 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))
3	1, 2	syl5 34	. 2 ⊢ (∃𝑥𝜑 → (𝜓 → ∃𝑥𝜓))
4		ax5e 1919	. 2 ⊢ (∃𝑥𝜓 → 𝜓)
5	3, 4	impbid1 226	1 ⊢ (∃𝑥𝜑 → (𝜓 ↔ ∃𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917

This theorem depends on definitions:  df-bi 208  df-ex 1787

This theorem is referenced by:  bj-exextruan  36979  bj-cbvexvv  36981