Theorem spfalw 2009

Description: Version of sp 2218 when 𝜑 is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.)

Hypothesis

Ref	Expression
spfalw.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
spfalw	⊢ (∀𝑥𝜑 → 𝜑)

Proof of Theorem spfalw

Step	Hyp	Ref	Expression
1		spfalw.1	. . 3 ⊢ ¬ 𝜑
2	1	hbth 1832	. 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
3	2	spnfw 2008	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:  ax6dgen  2162  axnulALT2  35480