Theorem spnfw 2008

Description: Weak version of sp 2218. 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 25	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜑))
3	1, 2	spimw 1999	1 ⊢ (∀𝑥𝜑 → 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-6 1996

This proof depends on definitions:  df-bi 210  df-ex 1809

This theorem is used by:  spfalw  2009  spvw  2010