Theorem spvw 2008

Description: Version of sp 2225 when 𝑥 does not occur in 𝜑. Converse of ax-5 1937. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) Shorten 19.3v 2009. (Revised by Wolf Lammen, 20-Oct-2023.)

Assertion

Ref	Expression
spvw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable group:   𝜑,𝑥

Proof of Theorem spvw

Step	Hyp	Ref	Expression
1		ax-5 1937	. 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
2	1	spnfw 2006	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1565

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  ax-6 1994

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

This theorem is referenced by:  19.3v  2009