MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  spfalw Structured version   Visualization version   GIF version

Theorem spfalw 1975
Description: Version of sp 2091 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 (∀𝑥𝜑𝜑)

Proof of Theorem spfalw
StepHypRef Expression
1 spfalw.1 . . 3 ¬ 𝜑
21hbth 1769 . 2 𝜑 → ∀𝑥 ¬ 𝜑)
32spnfw 1974 1 (∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-6 1945
This theorem depends on definitions:  df-bi 197  df-ex 1745
This theorem is referenced by:  ax6dgen  2045
  Copyright terms: Public domain W3C validator