MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfnf1 Structured version   Visualization version   GIF version

Theorem nfnf1 2028
Description: The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2044. (Revised by Wolf Lammen, 12-Oct-2021.)
Assertion
Ref Expression
nfnf1 𝑥𝑥𝜑

Proof of Theorem nfnf1
StepHypRef Expression
1 df-nf 1707 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2 nfe1 2024 . . 3 𝑥𝑥𝜑
3 nfa1 2025 . . 3 𝑥𝑥𝜑
42, 3nfim 1822 . 2 𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
51, 4nfxfr 1776 1 𝑥𝑥𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wex 1701  wnf 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-10 2016
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  nfaldOLD  2163  nfeqf2  2296  nfsb4t  2388  nfnfc1  2764  sbcnestgf  3967  dfnfc2OLD  4421  bj-sbf4  32467  wl-equsal1t  32956  wl-sb6rft  32959  wl-sb8t  32962  wl-mo2tf  32982  wl-eutf  32984  wl-mo2t  32986  wl-mo3t  32987  wl-sb8eut  32988
  Copyright terms: Public domain W3C validator