Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  axext4dist Structured version   Unicode version

Theorem axext4dist 29398
Description: axext4 2364 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)
Assertion
Ref Expression
axext4dist  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  <->  A. z ( z  e.  x  <->  z  e.  y ) ) )

Proof of Theorem axext4dist
StepHypRef Expression
1 axc9 2052 . . . 4  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
21imp 427 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  A. z  x  =  y )
)
3 nfnae 2064 . . . . 5  |-  F/ z  -.  A. z  z  =  x
4 nfnae 2064 . . . . 5  |-  F/ z  -.  A. z  z  =  y
53, 4nfan 1936 . . . 4  |-  F/ z ( -.  A. z 
z  =  x  /\  -.  A. z  z  =  y )
6 elequ2 1831 . . . . 5  |-  ( x  =  y  ->  (
z  e.  x  <->  z  e.  y ) )
76a1i 11 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  ( z  e.  x  <->  z  e.  y ) ) )
85, 7alimd 1884 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( A. z  x  =  y  ->  A. z ( z  e.  x  <->  z  e.  y ) ) )
92, 8syld 44 . 2  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  A. z
( z  e.  x  <->  z  e.  y ) ) )
10 axextdist 29397 . 2  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y ) )
119, 10impbid 191 1  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  <->  A. z ( z  e.  x  <->  z  e.  y ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-cleq 2374  df-clel 2377  df-nfc 2532
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator