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

Theorem equveli 2180
Description: A variable elimination law for equality with no distinct variable requirements. Compare equvini 2179. (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 albi 1690 . 2  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  ( A. z  z  =  x  <->  A. z  z  =  y ) )
2 ax6e 2094 . . . 4  |-  E. z 
z  =  y
3 biimpr 202 . . . . . 6  |-  ( ( z  =  x  <->  z  =  y )  ->  (
z  =  y  -> 
z  =  x ) )
4 ax7 1860 . . . . . 6  |-  ( z  =  x  ->  (
z  =  y  ->  x  =  y )
)
53, 4syli 38 . . . . 5  |-  ( ( z  =  x  <->  z  =  y )  ->  (
z  =  y  ->  x  =  y )
)
65com12 32 . . . 4  |-  ( z  =  y  ->  (
( z  =  x  <-> 
z  =  y )  ->  x  =  y ) )
72, 6eximii 1709 . . 3  |-  E. z
( ( z  =  x  <->  z  =  y )  ->  x  =  y )
8719.35i 1741 . 2  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  E. z  x  =  y )
94spsd 1945 . . . . 5  |-  ( z  =  x  ->  ( A. z  z  =  y  ->  x  =  y ) )
109sps 1943 . . . 4  |-  ( A. z  z  =  x  ->  ( A. z  z  =  y  ->  x  =  y ) )
1110a1dd 47 . . 3  |-  ( A. z  z  =  x  ->  ( A. z  z  =  y  ->  ( E. z  x  =  y  ->  x  =  y ) ) )
12 nfeqf 2139 . . . . 5  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
131219.9d 1968 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( E. z  x  =  y  ->  x  =  y ) )
1413ex 436 . . 3  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( E. z  x  =  y  ->  x  =  y ) ) )
1511, 14bija 357 . 2  |-  ( ( A. z  z  =  x  <->  A. z  z  =  y )  ->  ( E. z  x  =  y  ->  x  =  y ) )
161, 8, 15sylc 62 1  |-  ( A. z ( z  =  x  <->  z  =  y )  ->  x  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1442   E.wex 1663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933  ax-13 2091
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator