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Theorem hbequid 1451
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 1381, ax-8 1440, ax-12 1447, and ax-gen 1383. This shows that this can be proved without ax-9 1469, even though the theorem equid 1634 cannot be. A shorter proof using ax-9 1469 is obtainable from equid 1634 and hbth 1397.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.)
Assertion
Ref Expression
hbequid  |-  ( x  =  x  ->  A. y  x  =  x )

Proof of Theorem hbequid
StepHypRef Expression
1 ax12or 1448 . 2  |-  ( A. y  y  =  x  \/  ( A. y  y  =  x  \/  A. y ( x  =  x  ->  A. y  x  =  x )
) )
2 ax-8 1440 . . . . . 6  |-  ( y  =  x  ->  (
y  =  x  ->  x  =  x )
)
32pm2.43i 48 . . . . 5  |-  ( y  =  x  ->  x  =  x )
43alimi 1389 . . . 4  |-  ( A. y  y  =  x  ->  A. y  x  =  x )
54a1d 22 . . 3  |-  ( A. y  y  =  x  ->  ( x  =  x  ->  A. y  x  =  x ) )
6 ax-4 1445 . . . 4  |-  ( A. y ( x  =  x  ->  A. y  x  =  x )  ->  ( x  =  x  ->  A. y  x  =  x ) )
75, 6jaoi 671 . . 3  |-  ( ( A. y  y  =  x  \/  A. y
( x  =  x  ->  A. y  x  =  x ) )  -> 
( x  =  x  ->  A. y  x  =  x ) )
85, 7jaoi 671 . 2  |-  ( ( A. y  y  =  x  \/  ( A. y  y  =  x  \/  A. y ( x  =  x  ->  A. y  x  =  x )
) )  ->  (
x  =  x  ->  A. y  x  =  x ) )
91, 8ax-mp 7 1  |-  ( x  =  x  ->  A. y  x  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 664   A.wal 1287    = wceq 1289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-gen 1383  ax-8 1440  ax-i12 1443  ax-4 1445
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  equveli  1689
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