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Theorem ax10-16 2142
Description: This theorem shows that, given ax-16 2096, we can derive a version of ax-10 2092. However, it is weaker than ax-10 2092 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
ax10-16  |-  ( A. x  x  =  z  ->  A. z  z  =  x )
Distinct variable group:    x, z

Proof of Theorem ax10-16
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax-16 2096 . . . 4  |-  ( A. x  x  =  z  ->  ( x  =  w  ->  A. x  x  =  w ) )
21alrimiv 1621 . . 3  |-  ( A. x  x  =  z  ->  A. w ( x  =  w  ->  A. x  x  =  w )
)
32a5i-o 2102 . 2  |-  ( A. x  x  =  z  ->  A. x A. w
( x  =  w  ->  A. x  x  =  w ) )
4 equequ1 1667 . . . . . 6  |-  ( x  =  z  ->  (
x  =  w  <->  z  =  w ) )
54cbvalv 1955 . . . . . . 7  |-  ( A. x  x  =  w  <->  A. z  z  =  w )
65a1i 10 . . . . . 6  |-  ( x  =  z  ->  ( A. x  x  =  w 
<-> 
A. z  z  =  w ) )
74, 6imbi12d 311 . . . . 5  |-  ( x  =  z  ->  (
( x  =  w  ->  A. x  x  =  w )  <->  ( z  =  w  ->  A. z 
z  =  w ) ) )
87albidv 1615 . . . 4  |-  ( x  =  z  ->  ( A. w ( x  =  w  ->  A. x  x  =  w )  <->  A. w ( z  =  w  ->  A. z 
z  =  w ) ) )
98cbvalv 1955 . . 3  |-  ( A. x A. w ( x  =  w  ->  A. x  x  =  w )  <->  A. z A. w ( z  =  w  ->  A. z  z  =  w ) )
109biimpi 186 . 2  |-  ( A. x A. w ( x  =  w  ->  A. x  x  =  w )  ->  A. z A. w
( z  =  w  ->  A. z  z  =  w ) )
11 nfa1-o 2118 . . . . . . 7  |-  F/ z A. z  z  =  w
121119.23 1809 . . . . . 6  |-  ( A. z ( z  =  w  ->  A. z 
z  =  w )  <-> 
( E. z  z  =  w  ->  A. z 
z  =  w ) )
1312albii 1556 . . . . 5  |-  ( A. w A. z ( z  =  w  ->  A. z 
z  =  w )  <->  A. w ( E. z 
z  =  w  ->  A. z  z  =  w ) )
14 a9ev 1646 . . . . . . . 8  |-  E. z 
z  =  w
15 pm2.27 35 . . . . . . . 8  |-  ( E. z  z  =  w  ->  ( ( E. z  z  =  w  ->  A. z  z  =  w )  ->  A. z 
z  =  w ) )
1614, 15ax-mp 8 . . . . . . 7  |-  ( ( E. z  z  =  w  ->  A. z 
z  =  w )  ->  A. z  z  =  w )
1716alimi 1549 . . . . . 6  |-  ( A. w ( E. z 
z  =  w  ->  A. z  z  =  w )  ->  A. w A. z  z  =  w )
18 equequ2 1669 . . . . . . . . 9  |-  ( w  =  x  ->  (
z  =  w  <->  z  =  x ) )
1918spv 1951 . . . . . . . 8  |-  ( A. w  z  =  w  ->  z  =  x )
2019sps-o 2111 . . . . . . 7  |-  ( A. z A. w  z  =  w  ->  z  =  x )
2120a7s 1721 . . . . . 6  |-  ( A. w A. z  z  =  w  ->  z  =  x )
2217, 21syl 15 . . . . 5  |-  ( A. w ( E. z 
z  =  w  ->  A. z  z  =  w )  ->  z  =  x )
2313, 22sylbi 187 . . . 4  |-  ( A. w A. z ( z  =  w  ->  A. z 
z  =  w )  ->  z  =  x )
2423a7s 1721 . . 3  |-  ( A. z A. w ( z  =  w  ->  A. z 
z  =  w )  ->  z  =  x )
2524a5i-o 2102 . 2  |-  ( A. z A. w ( z  =  w  ->  A. z 
z  =  w )  ->  A. z  z  =  x )
263, 10, 253syl 18 1  |-  ( A. x  x  =  z  ->  A. z  z  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530   E.wex 1531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-4 2087  ax-5o 2088  ax-6o 2089  ax-16 2096
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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