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Theorem hbequid 2099
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 2077.) (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 2081 . 2  |-  ( -. 
A. y  y  =  x  ->  ( -.  A. y  y  =  x  ->  ( x  =  x  ->  A. y  x  =  x )
) )
2 ax-8 1643 . . . . 5  |-  ( y  =  x  ->  (
y  =  x  ->  x  =  x )
)
32pm2.43i 43 . . . 4  |-  ( y  =  x  ->  x  =  x )
43alimi 1546 . . 3  |-  ( A. y  y  =  x  ->  A. y  x  =  x )
54a1d 22 . 2  |-  ( A. y  y  =  x  ->  ( x  =  x  ->  A. y  x  =  x ) )
61, 5, 5pm2.61ii 157 1  |-  ( x  =  x  ->  A. y  x  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527
This theorem is referenced by:  nfequid-o  2100  equidq  2114
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
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