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

Theorem axextnd 9047
Description: A version of the Axiom of Extensionality with no distinct variable conditions. (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
axextnd  |-  E. x
( ( x  e.  y  <->  x  e.  z
)  ->  y  =  z )

Proof of Theorem axextnd
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfnae 2163 . . . . . . . 8  |-  F/ x  -.  A. x  x  =  y
2 nfnae 2163 . . . . . . . 8  |-  F/ x  -.  A. x  x  =  z
31, 2nfan 2022 . . . . . . 7  |-  F/ x
( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )
4 nfcvf 2626 . . . . . . . . . 10  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
54adantr 471 . . . . . . . . 9  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  F/_ x y )
65nfcrd 2609 . . . . . . . 8  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  F/ x  w  e.  y )
7 nfcvf 2626 . . . . . . . . . 10  |-  ( -. 
A. x  x  =  z  ->  F/_ x z )
87adantl 472 . . . . . . . . 9  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  F/_ x z )
98nfcrd 2609 . . . . . . . 8  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  F/ x  w  e.  z )
106, 9nfbid 2027 . . . . . . 7  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  F/ x
( w  e.  y  <-> 
w  e.  z ) )
11 elequ1 1905 . . . . . . . . 9  |-  ( w  =  x  ->  (
w  e.  y  <->  x  e.  y ) )
12 elequ1 1905 . . . . . . . . 9  |-  ( w  =  x  ->  (
w  e.  z  <->  x  e.  z ) )
1311, 12bibi12d 327 . . . . . . . 8  |-  ( w  =  x  ->  (
( w  e.  y  <-> 
w  e.  z )  <-> 
( x  e.  y  <-> 
x  e.  z ) ) )
1413a1i 11 . . . . . . 7  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  ( w  =  x  ->  ( ( w  e.  y  <->  w  e.  z )  <->  ( x  e.  y  <->  x  e.  z
) ) ) )
153, 10, 14cbvald 2129 . . . . . 6  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  ( A. w ( w  e.  y  <->  w  e.  z
)  <->  A. x ( x  e.  y  <->  x  e.  z ) ) )
16 axext3 2444 . . . . . 6  |-  ( A. w ( w  e.  y  <->  w  e.  z
)  ->  y  =  z )
1715, 16syl6bir 237 . . . . 5  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  ( A. x ( x  e.  y  <->  x  e.  z
)  ->  y  =  z ) )
18 19.8a 1946 . . . . 5  |-  ( y  =  z  ->  E. x  y  =  z )
1917, 18syl6 34 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  ( A. x ( x  e.  y  <->  x  e.  z
)  ->  E. x  y  =  z )
)
2019ex 440 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( A. x
( x  e.  y  <-> 
x  e.  z )  ->  E. x  y  =  z ) ) )
21 ax6e 2105 . . . . 5  |-  E. x  x  =  z
22 ax7 1871 . . . . . 6  |-  ( x  =  y  ->  (
x  =  z  -> 
y  =  z ) )
2322aleximi 1715 . . . . 5  |-  ( A. x  x  =  y  ->  ( E. x  x  =  z  ->  E. x  y  =  z )
)
2421, 23mpi 20 . . . 4  |-  ( A. x  x  =  y  ->  E. x  y  =  z )
2524a1d 26 . . 3  |-  ( A. x  x  =  y  ->  ( A. x ( x  e.  y  <->  x  e.  z )  ->  E. x  y  =  z )
)
26 ax6e 2105 . . . . 5  |-  E. x  x  =  y
27 ax7 1871 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
28 equcomi 1872 . . . . . . 7  |-  ( z  =  y  ->  y  =  z )
2927, 28syl6 34 . . . . . 6  |-  ( x  =  z  ->  (
x  =  y  -> 
y  =  z ) )
3029aleximi 1715 . . . . 5  |-  ( A. x  x  =  z  ->  ( E. x  x  =  y  ->  E. x  y  =  z )
)
3126, 30mpi 20 . . . 4  |-  ( A. x  x  =  z  ->  E. x  y  =  z )
3231a1d 26 . . 3  |-  ( A. x  x  =  z  ->  ( A. x ( x  e.  y  <->  x  e.  z )  ->  E. x  y  =  z )
)
3320, 25, 32pm2.61ii 170 . 2  |-  ( A. x ( x  e.  y  <->  x  e.  z
)  ->  E. x  y  =  z )
343319.35ri 1753 1  |-  E. x
( ( x  e.  y  <->  x  e.  z
)  ->  y  =  z )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1453   E.wex 1674   F/_wnfc 2590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-cleq 2455  df-clel 2458  df-nfc 2592
This theorem is referenced by:  zfcndext  9069  axextprim  30378  axextdfeq  30494  axextndbi  30501
  Copyright terms: Public domain W3C validator