Theorem spfalw 1672

Description: Version of sp 1747 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 1552	. 2 ⊢ (¬ φ → ∀x ¬ φ)
3	2	spnfw 1670	1 ⊢ (∀xφ → φ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-9 1654

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  19.2OLD  1700  ax9dgen  1716