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Theorem equveli 2081
Description: A variable elimination law for equality with no distinct variable requirements. (Compare equvini 2079.) (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 15-Apr-2018.)
Assertion
Ref Expression
equveli  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  x  =  y )

Proof of Theorem equveli
StepHypRef Expression
1 nfeqf 2009 . . . . 5  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
2 a9e 1952 . . . . . . 7  |-  E. z 
z  =  y
3 bi2 190 . . . . . . . . 9  |-  ( ( z  =  x  <->  z  =  y )  ->  (
z  =  y  -> 
z  =  x ) )
4 ax-8 1687 . . . . . . . . 9  |-  ( z  =  x  ->  (
z  =  y  ->  x  =  y )
)
53, 4syl6com 33 . . . . . . . 8  |-  ( z  =  y  ->  (
( z  =  x  <-> 
z  =  y )  ->  ( z  =  y  ->  x  =  y ) ) )
65pm2.43a 47 . . . . . . 7  |-  ( z  =  y  ->  (
( z  =  x  <-> 
z  =  y )  ->  x  =  y ) )
72, 6eximii 1587 . . . . . 6  |-  E. z
( ( z  =  x  <->  z  =  y )  ->  x  =  y )
8719.35i 1611 . . . . 5  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  E. z  x  =  y )
9 nf2 1889 . . . . . 6  |-  ( F/ z  x  =  y  <-> 
( E. z  x  =  y  ->  A. z  x  =  y )
)
109biimpi 187 . . . . 5  |-  ( F/ z  x  =  y  ->  ( E. z  x  =  y  ->  A. z  x  =  y ) )
111, 8, 10syl2im 36 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( A. z ( z  =  x  <->  z  =  y )  ->  A. z  x  =  y )
)
1211ex 424 . . 3  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( A. z
( z  =  x  <-> 
z  =  y )  ->  A. z  x  =  y ) ) )
13 bi1 179 . . . . . 6  |-  ( ( z  =  x  <->  z  =  y )  ->  (
z  =  x  -> 
z  =  y ) )
144com12 29 . . . . . 6  |-  ( z  =  y  ->  (
z  =  x  ->  x  =  y )
)
1513, 14syl6com 33 . . . . 5  |-  ( z  =  x  ->  (
( z  =  x  <-> 
z  =  y )  ->  ( z  =  x  ->  x  =  y ) ) )
1615pm2.43a 47 . . . 4  |-  ( z  =  x  ->  (
( z  =  x  <-> 
z  =  y )  ->  x  =  y ) )
1716al2imi 1570 . . 3  |-  ( A. z  z  =  x  ->  ( A. z ( z  =  x  <->  z  =  y )  ->  A. z  x  =  y )
)
186al2imi 1570 . . 3  |-  ( A. z  z  =  y  ->  ( A. z ( z  =  x  <->  z  =  y )  ->  A. z  x  =  y )
)
1912, 17, 18pm2.61ii 159 . 2  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  A. z  x  =  y )
201919.21bi 1774 1  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550   F/wnf 1553
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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