Theorem spnfw 1988

Description: Weak version of sp 2182. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 13-Aug-2017.)

Hypothesis

Ref	Expression
spnfw.1	⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)

Assertion

Ref	Expression
spnfw	⊢ (∀𝑥𝜑 → 𝜑)

Proof of Theorem spnfw

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		spnfw.1	. 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
2		idd 24	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜑))
3	1, 2	spimw 1979	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-6 1976

This theorem depends on definitions:  df-bi 210  df-ex 1788

This theorem is referenced by:  spvw  1989  spfalw  2006