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Theorem nfequid-o 2100
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-5 1544, ax-8 1643, ax-12o 2081, and ax-gen 1533. This shows that this can be proved without ax9 1889, even though the theorem equid 1644 cannot be. A shorter proof using ax9 1889 is obtainable from equid 1644 and hbth 1539.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax9v 1636, which is used for the derivation of ax12o 1875, unless we consider ax-12o 2081 the starting axiom rather than ax-12 1866. (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  |-  F/ y  x  =  x

Proof of Theorem nfequid-o
StepHypRef Expression
1 hbequid 2099 . 2  |-  ( x  =  x  ->  A. y  x  =  x )
21nfi 1538 1  |-  F/ y  x  =  x
Colors of variables: wff set class
Syntax hints:   F/wnf 1531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-8 1643  ax-12o 2081
This theorem depends on definitions:  df-bi 177  df-nf 1532
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