Theorem bj-spvw 37142

Description: Version of spvw 2008 and 19.3v 2009 proved from ax-1 6-- ax-5 1937. The antecedent can for instance be proved with the existence axiom extru 2002. (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 1937	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2		bj-axdd2 37070	. . 3 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))
3		ax5e 1939	. . 3 ⊢ (∃𝑥𝜓 → 𝜓)
4	2, 3	syl6 36	. 2 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜓))
5	1, 4	impbid2 229	1 ⊢ (∃𝑥𝜑 → (𝜓 ↔ ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565  ∃wex 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937

This theorem depends on definitions:  df-bi 210  df-ex 1807

This theorem is referenced by:  bj-alextruim  37144  bj-cbvalvv  37146  bj-axnul  37592