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Theorem axc11n-16 2261
Description: This theorem shows that, given ax-c16 2216, we can derive a version of ax-c11n 2212. However, it is weaker than ax-c11n 2212 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc11n-16  |-  ( A. x  x  =  z  ->  A. z  z  =  x )
Distinct variable group:    x, z

Proof of Theorem axc11n-16
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax-c16 2216 . . . 4  |-  ( A. x  x  =  z  ->  ( x  =  w  ->  A. x  x  =  w ) )
21alrimiv 1695 . . 3  |-  ( A. x  x  =  z  ->  A. w ( x  =  w  ->  A. x  x  =  w )
)
32axc4i-o 2222 . 2  |-  ( A. x  x  =  z  ->  A. x A. w
( x  =  w  ->  A. x  x  =  w ) )
4 equequ1 1747 . . . . . 6  |-  ( x  =  z  ->  (
x  =  w  <->  z  =  w ) )
54cbvalv 1996 . . . . . . 7  |-  ( A. x  x  =  w  <->  A. z  z  =  w )
65a1i 11 . . . . . 6  |-  ( x  =  z  ->  ( A. x  x  =  w 
<-> 
A. z  z  =  w ) )
74, 6imbi12d 320 . . . . 5  |-  ( x  =  z  ->  (
( x  =  w  ->  A. x  x  =  w )  <->  ( z  =  w  ->  A. z 
z  =  w ) ) )
87albidv 1689 . . . 4  |-  ( x  =  z  ->  ( A. w ( x  =  w  ->  A. x  x  =  w )  <->  A. w ( z  =  w  ->  A. z 
z  =  w ) ) )
98cbvalv 1996 . . 3  |-  ( A. x A. w ( x  =  w  ->  A. x  x  =  w )  <->  A. z A. w ( z  =  w  ->  A. z  z  =  w ) )
109biimpi 194 . 2  |-  ( A. x A. w ( x  =  w  ->  A. x  x  =  w )  ->  A. z A. w
( z  =  w  ->  A. z  z  =  w ) )
11 nfa1-o 2238 . . . . . . 7  |-  F/ z A. z  z  =  w
121119.23 1857 . . . . . 6  |-  ( A. z ( z  =  w  ->  A. z 
z  =  w )  <-> 
( E. z  z  =  w  ->  A. z 
z  =  w ) )
1312albii 1620 . . . . 5  |-  ( A. w A. z ( z  =  w  ->  A. z 
z  =  w )  <->  A. w ( E. z 
z  =  w  ->  A. z  z  =  w ) )
14 ax6ev 1721 . . . . . . . 8  |-  E. z 
z  =  w
15 pm2.27 39 . . . . . . . 8  |-  ( E. z  z  =  w  ->  ( ( E. z  z  =  w  ->  A. z  z  =  w )  ->  A. z 
z  =  w ) )
1614, 15ax-mp 5 . . . . . . 7  |-  ( ( E. z  z  =  w  ->  A. z 
z  =  w )  ->  A. z  z  =  w )
1716alimi 1614 . . . . . 6  |-  ( A. w ( E. z 
z  =  w  ->  A. z  z  =  w )  ->  A. w A. z  z  =  w )
18 equequ2 1748 . . . . . . . . 9  |-  ( w  =  x  ->  (
z  =  w  <->  z  =  x ) )
1918spv 1980 . . . . . . . 8  |-  ( A. w  z  =  w  ->  z  =  x )
2019sps-o 2231 . . . . . . 7  |-  ( A. z A. w  z  =  w  ->  z  =  x )
2120alcoms 1792 . . . . . 6  |-  ( A. w A. z  z  =  w  ->  z  =  x )
2217, 21syl 16 . . . . 5  |-  ( A. w ( E. z 
z  =  w  ->  A. z  z  =  w )  ->  z  =  x )
2313, 22sylbi 195 . . . 4  |-  ( A. w A. z ( z  =  w  ->  A. z 
z  =  w )  ->  z  =  x )
2423alcoms 1792 . . 3  |-  ( A. z A. w ( z  =  w  ->  A. z 
z  =  w )  ->  z  =  x )
2524axc4i-o 2222 . 2  |-  ( A. z A. w ( z  =  w  ->  A. z 
z  =  w )  ->  A. z  z  =  x )
263, 10, 253syl 20 1  |-  ( A. x  x  =  z  ->  A. z  z  =  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377   E.wex 1596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-c5 2207  ax-c4 2208  ax-c7 2209  ax-c16 2216
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600
This theorem is referenced by: (None)
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