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Theorem hbequid 1422
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 1352, ax-8 1411, ax-12 1418, and ax-gen 1354. This shows that this can be proved without ax-9 1440, even though the theorem equid 1605 cannot be. A shorter proof using ax-9 1440 is obtainable from equid 1605 and hbth 1368.) (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 1419 . 2  |-  ( A. y  y  =  x  \/  ( A. y  y  =  x  \/  A. y ( x  =  x  ->  A. y  x  =  x )
) )
2 ax-8 1411 . . . . . 6  |-  ( y  =  x  ->  (
y  =  x  ->  x  =  x )
)
32pm2.43i 47 . . . . 5  |-  ( y  =  x  ->  x  =  x )
43alimi 1360 . . . 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 1416 . . . 4  |-  ( A. y ( x  =  x  ->  A. y  x  =  x )  ->  ( x  =  x  ->  A. y  x  =  x ) )
75, 6jaoi 646 . . 3  |-  ( ( A. y  y  =  x  \/  A. y
( x  =  x  ->  A. y  x  =  x ) )  -> 
( x  =  x  ->  A. y  x  =  x ) )
85, 7jaoi 646 . 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 639   A.wal 1257    = wceq 1259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-gen 1354  ax-8 1411  ax-i12 1414  ax-4 1416
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  equveli  1658
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