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

Theorem nf5r 2062
 Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1707 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
Assertion
Ref Expression
nf5r (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Proof of Theorem nf5r
StepHypRef Expression
1 19.8a 2049 . 2 (𝜑 → ∃𝑥𝜑)
2 df-nf 1707 . . 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 1478  ∃wex 1701  Ⅎwnf 1705 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044 This theorem depends on definitions:  df-bi 197  df-ex 1702  df-nf 1707 This theorem is referenced by:  nf5ri  2063  nf5rd  2064  19.21tOLDOLD  2072  sbft  2378  bj-alrim  32378  bj-nexdt  32382  bj-cbv3tb  32406  bj-nfs1t2  32410  bj-sbftv  32459  bj-equsal1t  32505  stdpc5t  32510  bj-axc14  32537  wl-nfeqfb  32994
 Copyright terms: Public domain W3C validator