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

Theorem nfnfc 2803
Description: Hypothesis builder for 𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2282. (Revised by Wolf Lammen, 10-Dec-2019.)
Hypothesis
Ref Expression
nfnfc.1 𝑥𝐴
Assertion
Ref Expression
nfnfc 𝑥𝑦𝐴

Proof of Theorem nfnfc
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-nfc 2782 . 2 (𝑦𝐴 ↔ ∀𝑧𝑦 𝑧𝐴)
2 nfnfc.1 . . . . 5 𝑥𝐴
3 nfcr 2785 . . . . 5 (𝑥𝐴 → Ⅎ𝑥 𝑧𝐴)
42, 3ax-mp 5 . . . 4 𝑥 𝑧𝐴
54nfnf 2196 . . 3 𝑥𝑦 𝑧𝐴
65nfal 2191 . 2 𝑥𝑧𝑦 𝑧𝐴
71, 6nfxfr 1819 1 𝑥𝑦𝐴
Colors of variables: wff setvar class
Syntax hints:  wal 1521  wnf 1748  wcel 2030  wnfc 2780
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-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-nfc 2782
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator