Theorem bj-spvw 37274

Description: Version of spvw 1982 proved from ax-1 6-- ax-5 1911 and the existence axiom extru 1976. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)

Assertion

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

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-spvw

Step	Hyp	Ref	Expression
1		bj-axdd2 36792	. 2 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))
2		ax5e 1913	. 2 ⊢ (∃𝑥𝜓 → 𝜓)
3	1, 2	syl6 35	1 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1539  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 207  df-ex 1781

This theorem is referenced by:  bj-axnul  37276