MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfequid-o Unicode version

Theorem nfequid-o 2240
Description: Bound-variable hypothesis builder for . This theorem tells us that any variable, including , is effectively not free in , even though is technically free according to the traditional definition of free variable. (The proof uses only ax-4 1631, ax-7 1790, ax-c9 2221, and ax-gen 1618. This shows that this can be proved without ax6 2003, even though the theorem equid 1791 cannot be. A shorter proof using ax6 2003 is obtainable from equid 1791 and hbth 1624.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1748, which is used for the derivation of axc9 2046, unless we consider ax-c9 2221 the starting axiom rather than ax-13 1999. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Ref Expression

Proof of Theorem nfequid-o
StepHypRef Expression
1 hbequid 2239 . 2
21nfi 1623 1
Colors of variables: wff setvar class
Syntax hints:  F/wnf 1616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-7 1790  ax-c9 2221
This theorem depends on definitions:  df-bi 185  df-nf 1617
  Copyright terms: Public domain W3C validator