Theorem spnfw 1670

Description: Weak version of sp 1747. 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 21	. 2 ⊢ (x = y → (φ → φ))
3	1, 2	spimw 1668	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:  spfalw  1672  19.8wOLD  1693