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Theorem hbae 2149
Description: All variables are effectively bound in an identical variable specifier. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)
Assertion
Ref Expression
hbae  |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )

Proof of Theorem hbae
StepHypRef Expression
1 sp 1937 . . . . 5  |-  ( A. x  x  =  y  ->  x  =  y )
2 axc9 2140 . . . . 5  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
31, 2syl7 70 . . . 4  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( A. x  x  =  y  ->  A. z  x  =  y ) ) )
4 axc112 2020 . . . 4  |-  ( A. z  z  =  x  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
5 axc11 2148 . . . . . 6  |-  ( A. x  x  =  y  ->  ( A. x  x  =  y  ->  A. y  x  =  y )
)
65pm2.43i 49 . . . . 5  |-  ( A. x  x  =  y  ->  A. y  x  =  y )
7 axc112 2020 . . . . 5  |-  ( A. z  z  =  y  ->  ( A. y  x  =  y  ->  A. z  x  =  y )
)
86, 7syl5 33 . . . 4  |-  ( A. z  z  =  y  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
93, 4, 8pm2.61ii 169 . . 3  |-  ( A. x  x  =  y  ->  A. z  x  =  y )
109axc4i 1980 . 2  |-  ( A. x  x  =  y  ->  A. x A. z  x  =  y )
11 ax-11 1920 . 2  |-  ( A. x A. z  x  =  y  ->  A. z A. x  x  =  y )
1210, 11syl 17 1  |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668
This theorem is referenced by:  nfae  2150  hbnae  2151  aevALT  2155  drex2  2162  ax6e2eq  36924
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