MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  equveli Structured version   Unicode version

Theorem equveli 2061
Description: A variable elimination law for equality with no distinct variable requirements. (Compare equvini 2060.) (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 15-Jun-2019.)
Assertion
Ref Expression
equveli  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  x  =  y )

Proof of Theorem equveli
StepHypRef Expression
1 ax6e 1971 . . . 4  |-  E. z 
z  =  y
2 bi2 198 . . . . . 6  |-  ( ( z  =  x  <->  z  =  y )  ->  (
z  =  y  -> 
z  =  x ) )
3 ax-7 1739 . . . . . 6  |-  ( z  =  x  ->  (
z  =  y  ->  x  =  y )
)
42, 3syli 37 . . . . 5  |-  ( ( z  =  x  <->  z  =  y )  ->  (
z  =  y  ->  x  =  y )
)
54com12 31 . . . 4  |-  ( z  =  y  ->  (
( z  =  x  <-> 
z  =  y )  ->  x  =  y ) )
61, 5eximii 1637 . . 3  |-  E. z
( ( z  =  x  <->  z  =  y )  ->  x  =  y )
7619.35i 1666 . 2  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  E. z  x  =  y )
8 albi 1619 . . 3  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  ( A. z  z  =  x  <->  A. z  z  =  y ) )
93spsd 1816 . . . . . 6  |-  ( z  =  x  ->  ( A. z  z  =  y  ->  x  =  y ) )
109sps 1814 . . . . 5  |-  ( A. z  z  =  x  ->  ( A. z  z  =  y  ->  x  =  y ) )
1110a1dd 46 . . . 4  |-  ( A. z  z  =  x  ->  ( A. z  z  =  y  ->  ( E. z  x  =  y  ->  x  =  y ) ) )
12 nfeqf 2018 . . . . . 6  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
131219.9d 1840 . . . . 5  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( E. z  x  =  y  ->  x  =  y ) )
1413ex 434 . . . 4  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( E. z  x  =  y  ->  x  =  y ) ) )
1511, 14bija 355 . . 3  |-  ( ( A. z  z  =  x  <->  A. z  z  =  y )  ->  ( E. z  x  =  y  ->  x  =  y ) )
168, 15syl 16 . 2  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  ( E. z  x  =  y  ->  x  =  y ) )
177, 16mpd 15 1  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  x  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   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-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator