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Theorem hbequid 1687
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 1533, ax-8 1623, ax-12o 1664, and ax-gen 1536. This shows that this can be proved without ax-9 1684, even though the theorem equid 1818 cannot be. A shorter proof using ax-9 1684 is obtainable from equid 1818 and hbth 1557.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax-9v 1632, which is used for the derivation of ax12o 1663, unless we consider ax-12o 1664 the starting axiom rather than ax-12 1633. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.)
Assertion
Ref Expression
hbequid  |-  ( x  =  x  ->  A. y  x  =  x )

Proof of Theorem hbequid
StepHypRef Expression
1 ax-12o 1664 . 2  |-  ( -. 
A. y  y  =  x  ->  ( -.  A. y  y  =  x  ->  ( x  =  x  ->  A. y  x  =  x )
) )
2 ax-8 1623 . . . . 5  |-  ( y  =  x  ->  (
y  =  x  ->  x  =  x )
)
32pm2.43i 45 . . . 4  |-  ( y  =  x  ->  x  =  x )
43alimi 1546 . . 3  |-  ( A. y  y  =  x  ->  A. y  x  =  x )
54a1d 24 . 2  |-  ( A. y  y  =  x  ->  ( x  =  x  ->  A. y  x  =  x ) )
61, 5, 5pm2.61ii 159 1  |-  ( x  =  x  ->  A. y  x  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532
This theorem is referenced by:  nfequid  1688  equidq  1689
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-8 1623  ax-12o 1664
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