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Theorem hbae-o 2092
Description: All variables are effectively bound in an identical variable specifier. Version of hbae 1893 using ax-10o 2078. (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 2074 . . . . 5  |-  ( A. x  x  =  y  ->  x  =  y )
2 ax-12o 2081 . . . . 5  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
31, 2syl7 63 . . . 4  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( A. x  x  =  y  ->  A. z  x  =  y ) ) )
4 ax-10o 2078 . . . . 5  |-  ( A. x  x  =  z  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
54aecoms-o 2091 . . . 4  |-  ( A. z  z  =  x  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
6 ax-10o 2078 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( A. x  x  =  y  ->  A. y  x  =  y )
)
76pm2.43i 43 . . . . . 6  |-  ( A. x  x  =  y  ->  A. y  x  =  y )
8 ax-10o 2078 . . . . . 6  |-  ( A. y  y  =  z  ->  ( A. y  x  =  y  ->  A. z  x  =  y )
)
97, 8syl5 28 . . . . 5  |-  ( A. y  y  =  z  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
109aecoms-o 2091 . . . 4  |-  ( A. z  z  =  y  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
113, 5, 10pm2.61ii 157 . . 3  |-  ( A. x  x  =  y  ->  A. z  x  =  y )
1211a5i-o 2089 . 2  |-  ( A. x  x  =  y  ->  A. x A. z  x  =  y )
13 ax-7 1708 . 2  |-  ( A. x A. z  x  =  y  ->  A. z A. x  x  =  y )
1412, 13syl 15 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 1527
This theorem is referenced by:  dral1-o  2093  hbnae-o  2118  dral2-o  2120  aev-o  2121
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-17 1603  ax-9 1635  ax-8 1643  ax-7 1708  ax-4 2074  ax-5o 2075  ax-6o 2076  ax-10o 2078  ax-12o 2081
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