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Theorem hbequid 2237
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 does not use ax-9o 2215.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
hbequid  |-  ( x  =  x  ->  A. y  x  =  x )

Proof of Theorem hbequid
StepHypRef Expression
1 ax-12o 2219 . 2  |-  ( -. 
A. y  y  =  x  ->  ( -.  A. y  y  =  x  ->  ( x  =  x  ->  A. y  x  =  x )
) )
2 ax-8 1687 . . . . 5  |-  ( y  =  x  ->  (
y  =  x  ->  x  =  x )
)
32pm2.43i 45 . . . 4  |-  ( y  =  x  ->  x  =  x )
43alimi 1568 . . 3  |-  ( A. y  y  =  x  ->  A. y  x  =  x )
54a1d 23 . 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 4   A.wal 1549
This theorem is referenced by:  nfequid-o  2238  equidq  2252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-8 1687  ax-12o 2219
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