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Theorem hbae-o 32181
Description: All variables are effectively bound in an identical variable specifier. Version of hbae 2111 using ax-c11 32167. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (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-c5 32163 . . . . 5  |-  ( A. x  x  =  y  ->  x  =  y )
2 ax-c9 32170 . . . . 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 ax-c11 32167 . . . . 5  |-  ( A. x  x  =  z  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
54aecoms-o 32180 . . . 4  |-  ( A. z  z  =  x  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
6 ax-c11 32167 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( A. x  x  =  y  ->  A. y  x  =  y )
)
76pm2.43i 49 . . . . . 6  |-  ( A. x  x  =  y  ->  A. y  x  =  y )
8 ax-c11 32167 . . . . . 6  |-  ( A. y  y  =  z  ->  ( A. y  x  =  y  ->  A. z  x  =  y )
)
97, 8syl5 33 . . . . 5  |-  ( A. y  y  =  z  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
109aecoms-o 32180 . . . 4  |-  ( A. z  z  =  y  ->  ( A. x  x  =  y  ->  A. z  x  =  y )
)
113, 5, 10pm2.61ii 168 . . 3  |-  ( A. x  x  =  y  ->  A. z  x  =  y )
1211axc4i-o 32178 . 2  |-  ( A. x  x  =  y  ->  A. x A. z  x  =  y )
13 ax-11 1894 . 2  |-  ( A. x A. z  x  =  y  ->  A. z A. x  x  =  y )
1412, 13syl 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 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-11 1894  ax-c5 32163  ax-c4 32164  ax-c7 32165  ax-c11 32167  ax-c9 32170
This theorem depends on definitions:  df-bi 188  df-ex 1660
This theorem is referenced by:  dral1-o  32182  hbnae-o  32207  dral2-o  32209  aev-o  32210
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