Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nfequid-o Structured version   Visualization version   GIF version

Theorem nfequid-o 33009
Description: Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (The proof uses only ax-4 1727, ax-7 1921, ax-c9 32989, and ax-gen 1712. This shows that this can be proved without ax6 2238, even though the theorem equid 1925 cannot be. A shorter proof using ax6 2238 is obtainable from equid 1925 and hbth 1719.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1875, which is used for the derivation of axc9 2289, unless we consider ax-c9 32989 the starting axiom rather than ax-13 2233. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Ref Expression
nfequid-o 𝑦 𝑥 = 𝑥

Proof of Theorem nfequid-o
StepHypRef Expression
1 hbequid 33008 . 2 (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
21nf5i 2010 1 𝑦 𝑥 = 𝑥
Colors of variables: wff setvar class
Syntax hints:  wnf 1698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-c9 32989
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-nf 1700
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator