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Theorem nfequid-o 32545
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 1690, ax-7 1859, ax-c9 32526, and ax-gen 1677. This shows that this can be proved without ax6 2108, even though the theorem equid 1863 cannot be. A shorter proof using ax6 2108 is obtainable from equid 1863 and hbth 1683.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1814, which is used for the derivation of axc9 2154, unless we consider ax-c9 32526 the starting axiom rather than ax-13 2104. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nfequid-o  |-  F/ y  x  =  x

Proof of Theorem nfequid-o
StepHypRef Expression
1 hbequid 32544 . 2  |-  ( x  =  x  ->  A. y  x  =  x )
21nfi 1682 1  |-  F/ y  x  =  x
Colors of variables: wff setvar class
Syntax hints:   F/wnf 1675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-c9 32526
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676
This theorem is referenced by: (None)
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