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Theorem axext4dist 24157
Description: axext4 2267 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 ax12o 1875 . . . 4  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
21imp 418 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  A. z  x  =  y )
)
3 nfnae 1896 . . . . 5  |-  F/ z  -.  A. z  z  =  x
4 nfnae 1896 . . . . 5  |-  F/ z  -.  A. z  z  =  y
53, 4nfan 1771 . . . 4  |-  F/ z ( -.  A. z 
z  =  x  /\  -.  A. z  z  =  y )
6 elequ2 1689 . . . . 5  |-  ( x  =  y  ->  (
z  e.  x  <->  z  e.  y ) )
76a1i 10 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  ( z  e.  x  <->  z  e.  y ) ) )
85, 7alimd 1744 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( A. z  x  =  y  ->  A. z ( z  e.  x  <->  z  e.  y ) ) )
92, 8syld 40 . 2  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  A. z
( z  e.  x  <->  z  e.  y ) ) )
10 axextdist 24156 . 2  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y ) )
119, 10impbid 183 1  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  <->  A. z ( z  e.  x  <->  z  e.  y ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408
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