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Theorem hbae-o 2230
Description: All variables are effectively bound in an identical variable specifier. Version of hbae 2040 using ax-10o 2216. (Contributed by NM, 5-Aug-1993.) (Proof modification is disccouraged.) (New usage is discouraged.)
Assertion
Ref Expression
hbae-o  |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )

Proof of Theorem hbae-o
StepHypRef Expression
1 ax-4 2212 . . . . 5  |-  ( A. x  x  =  y  ->  x  =  y )
2 ax-12o 2219 . . . . 5  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
31, 2syl7 65 . . . 4  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( A. x  x  =  y  ->  A. z  x  =  y ) ) )
4 ax-10o 2216 . . . . 5  |-  ( A. x  x  =  z  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
54aecoms-o 2229 . . . 4  |-  ( A. z  z  =  x  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
6 ax-10o 2216 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( A. x  x  =  y  ->  A. y  x  =  y )
)
76pm2.43i 45 . . . . . 6  |-  ( A. x  x  =  y  ->  A. y  x  =  y )
8 ax-10o 2216 . . . . . 6  |-  ( A. y  y  =  z  ->  ( A. y  x  =  y  ->  A. z  x  =  y )
)
97, 8syl5 30 . . . . 5  |-  ( A. y  y  =  z  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
109aecoms-o 2229 . . . 4  |-  ( A. z  z  =  y  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
113, 5, 10pm2.61ii 159 . . 3  |-  ( A. x  x  =  y  ->  A. z  x  =  y )
1211a5i-o 2227 . 2  |-  ( A. x  x  =  y  ->  A. x A. z  x  =  y )
13 ax-7 1749 . 2  |-  ( A. x A. z  x  =  y  ->  A. z A. x  x  =  y )
1412, 13syl 16 1  |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549
This theorem is referenced by:  dral1-o  2231  hbnae-o  2256  dral2-o  2258  aev-o  2259
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-17 1626  ax-9 1666  ax-8 1687  ax-7 1749  ax-4 2212  ax-5o 2213  ax-6o 2214  ax-10o 2216  ax-12o 2219
This theorem depends on definitions:  df-bi 178  df-ex 1551
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