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Theorem nfequid-o 2161
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 1557, ax-8 1675, ax-12o 2142, and ax-gen 1546. This shows that this can be proved without ax9 1949, even though the theorem equid 1676 cannot be. A shorter proof using ax9 1949 is obtainable from equid 1676 and hbth 1552.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax9v 1655, which is used for the derivation of ax12o 1934, unless we consider ax-12o 2142 the starting axiom rather than ax-12 1925. (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 2160 . 2 (x = xy x = x)
21nfi 1551 1 y x = x
Colors of variables: wff setvar class
Syntax hints:  wnf 1544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-8 1675  ax-12o 2142
This theorem depends on definitions:  df-bi 177  df-nf 1545
This theorem is referenced by: (None)
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