MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nf5r Structured version   Visualization version   GIF version

Theorem nf5r 2218
Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1858 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
Assertion
Ref Expression
nf5r (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Proof of Theorem nf5r
StepHypRef Expression
1 19.8a 2206 . 2 (𝜑 → ∃𝑥𝜑)
2 df-nf 1858 . . 3 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
32biimpi 206 . 2 (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
41, 3syl5 34 1 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629  wex 1852  wnf 1856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-ex 1853  df-nf 1858
This theorem is referenced by:  nf5ri  2219  nf5rd  2220  19.21tOLDOLD  2230  sbft  2526  bj-alrim  33020  bj-nexdt  33024  bj-cbv3tb  33048  bj-nfs1t2  33052  bj-sbftv  33099  bj-equsal1t  33144  stdpc5t  33149  bj-axc14  33173  wl-nfeqfb  33658
  Copyright terms: Public domain W3C validator