Theorem spvw 1981

Description: Version of sp 2178 when 𝑥 does not occur in 𝜑. Converse of ax-5 1907. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) Shorten 19.3v 1982. (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 1907	. 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
2	1	spnfw 1980	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966

This theorem depends on definitions:  df-bi 209  df-ex 1777

This theorem is referenced by:  19.3v  1982