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Theorem equveli 1941
Description: A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1940.) (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)
Assertion
Ref Expression
equveli  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  x  =  y )

Proof of Theorem equveli
StepHypRef Expression
1 albiim 1601 . 2  |-  ( A. z ( z  =  x  <->  z  =  y )  <->  ( A. z
( z  =  x  ->  z  =  y )  /\  A. z
( z  =  y  ->  z  =  x ) ) )
2 equequ1 1667 . . . . . . . 8  |-  ( z  =  y  ->  (
z  =  y  <->  y  =  y ) )
3 equequ1 1667 . . . . . . . 8  |-  ( z  =  y  ->  (
z  =  x  <->  y  =  x ) )
42, 3imbi12d 311 . . . . . . 7  |-  ( z  =  y  ->  (
( z  =  y  ->  z  =  x )  <->  ( y  =  y  ->  y  =  x ) ) )
54sps 1751 . . . . . 6  |-  ( A. z  z  =  y  ->  ( ( z  =  y  ->  z  =  x )  <->  ( y  =  y  ->  y  =  x ) ) )
65dral1 1918 . . . . 5  |-  ( A. z  z  =  y  ->  ( A. z ( z  =  y  -> 
z  =  x )  <->  A. y ( y  =  y  ->  y  =  x ) ) )
7 equid 1662 . . . . . . 7  |-  y  =  y
8 sp 1728 . . . . . . 7  |-  ( A. y ( y  =  y  ->  y  =  x )  ->  (
y  =  y  -> 
y  =  x ) )
97, 8mpi 16 . . . . . 6  |-  ( A. y ( y  =  y  ->  y  =  x )  ->  y  =  x )
10 equcomi 1664 . . . . . 6  |-  ( y  =  x  ->  x  =  y )
119, 10syl 15 . . . . 5  |-  ( A. y ( y  =  y  ->  y  =  x )  ->  x  =  y )
126, 11syl6bi 219 . . . 4  |-  ( A. z  z  =  y  ->  ( A. z ( z  =  y  -> 
z  =  x )  ->  x  =  y ) )
1312adantld 453 . . 3  |-  ( A. z  z  =  y  ->  ( ( A. z
( z  =  x  ->  z  =  y )  /\  A. z
( z  =  y  ->  z  =  x ) )  ->  x  =  y ) )
14 equequ1 1667 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z  =  x  <->  x  =  x ) )
15 equequ1 1667 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z  =  y  <->  x  =  y ) )
1614, 15imbi12d 311 . . . . . . . . 9  |-  ( z  =  x  ->  (
( z  =  x  ->  z  =  y )  <->  ( x  =  x  ->  x  =  y ) ) )
1716sps 1751 . . . . . . . 8  |-  ( A. z  z  =  x  ->  ( ( z  =  x  ->  z  =  y )  <->  ( x  =  x  ->  x  =  y ) ) )
1817dral2 1919 . . . . . . 7  |-  ( A. z  z  =  x  ->  ( A. z ( z  =  x  -> 
z  =  y )  <->  A. z ( x  =  x  ->  x  =  y ) ) )
19 equid 1662 . . . . . . . . . 10  |-  x  =  x
2019a1bi 327 . . . . . . . . 9  |-  ( x  =  y  <->  ( x  =  x  ->  x  =  y ) )
2120biimpri 197 . . . . . . . 8  |-  ( ( x  =  x  ->  x  =  y )  ->  x  =  y )
2221sps 1751 . . . . . . 7  |-  ( A. z ( x  =  x  ->  x  =  y )  ->  x  =  y )
2318, 22syl6bi 219 . . . . . 6  |-  ( A. z  z  =  x  ->  ( A. z ( z  =  x  -> 
z  =  y )  ->  x  =  y ) )
2423a1d 22 . . . . 5  |-  ( A. z  z  =  x  ->  ( -.  A. z 
z  =  y  -> 
( A. z ( z  =  x  -> 
z  =  y )  ->  x  =  y ) ) )
25 nfeqf 1911 . . . . . . 7  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
26 equtr 1671 . . . . . . . . . 10  |-  ( z  =  x  ->  (
x  =  x  -> 
z  =  x ) )
27 ax-8 1661 . . . . . . . . . 10  |-  ( z  =  x  ->  (
z  =  y  ->  x  =  y )
)
2826, 27imim12d 68 . . . . . . . . 9  |-  ( z  =  x  ->  (
( z  =  x  ->  z  =  y )  ->  ( x  =  x  ->  x  =  y ) ) )
2919, 28mpii 39 . . . . . . . 8  |-  ( z  =  x  ->  (
( z  =  x  ->  z  =  y )  ->  x  =  y ) )
3029ax-gen 1536 . . . . . . 7  |-  A. z
( z  =  x  ->  ( ( z  =  x  ->  z  =  y )  ->  x  =  y )
)
31 spimt 1927 . . . . . . 7  |-  ( ( F/ z  x  =  y  /\  A. z
( z  =  x  ->  ( ( z  =  x  ->  z  =  y )  ->  x  =  y )
) )  ->  ( A. z ( z  =  x  ->  z  =  y )  ->  x  =  y ) )
3225, 30, 31sylancl 643 . . . . . 6  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( A. z ( z  =  x  ->  z  =  y )  ->  x  =  y ) )
3332ex 423 . . . . 5  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( A. z
( z  =  x  ->  z  =  y )  ->  x  =  y ) ) )
3424, 33pm2.61i 156 . . . 4  |-  ( -. 
A. z  z  =  y  ->  ( A. z ( z  =  x  ->  z  =  y )  ->  x  =  y ) )
3534adantrd 454 . . 3  |-  ( -. 
A. z  z  =  y  ->  ( ( A. z ( z  =  x  ->  z  =  y )  /\  A. z ( z  =  y  ->  z  =  x ) )  ->  x  =  y )
)
3613, 35pm2.61i 156 . 2  |-  ( ( A. z ( z  =  x  ->  z  =  y )  /\  A. z ( z  =  y  ->  z  =  x ) )  ->  x  =  y )
371, 36sylbi 187 1  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   F/wnf 1534
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
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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