Theorem bj-spvw 37068

Description: Version of spvw 2000 and 19.3v 2001 proved from ax-1 6-- ax-5 1929. The antecedent can for instance be proved with the existence axiom extru 1994. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)

Assertion

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

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-spvw

Step	Hyp	Ref	Expression
1		ax-5 1929	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2		bj-axdd2 36996	. . 3 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))
3		ax5e 1931	. . 3 ⊢ (∃𝑥𝜓 → 𝜓)
4	2, 3	syl6 35	. 2 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜓))
5	1, 4	impbid2 228	1 ⊢ (∃𝑥𝜑 → (𝜓 ↔ ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1557  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929

This theorem depends on definitions:  df-bi 209  df-ex 1799

This theorem is referenced by:  bj-alextruim  37070  bj-cbvalvv  37072  bj-axnul  37518