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Theorem axextdist 24156
Description: ax-ext 2264 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)
Assertion
Ref Expression
axextdist  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y ) )

Proof of Theorem axextdist
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfnae 1896 . . . 4  |-  F/ z  -.  A. z  z  =  x
2 nfnae 1896 . . . 4  |-  F/ z  -.  A. z  z  =  y
31, 2nfan 1771 . . 3  |-  F/ z ( -.  A. z 
z  =  x  /\  -.  A. z  z  =  y )
4 nfcvd 2420 . . . . 5  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/_ z w )
5 nfcvf 2441 . . . . . 6  |-  ( -. 
A. z  z  =  x  ->  F/_ z x )
65adantr 451 . . . . 5  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/_ z x )
74, 6nfeld 2434 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  w  e.  x )
8 nfcvf 2441 . . . . . 6  |-  ( -. 
A. z  z  =  y  ->  F/_ z y )
98adantl 452 . . . . 5  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/_ z y )
104, 9nfeld 2434 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  w  e.  y )
117, 10nfbid 1762 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z
( w  e.  x  <->  w  e.  y ) )
12 elequ1 1687 . . . . 5  |-  ( w  =  z  ->  (
w  e.  x  <->  z  e.  x ) )
13 elequ1 1687 . . . . 5  |-  ( w  =  z  ->  (
w  e.  y  <->  z  e.  y ) )
1412, 13bibi12d 312 . . . 4  |-  ( w  =  z  ->  (
( w  e.  x  <->  w  e.  y )  <->  ( z  e.  x  <->  z  e.  y ) ) )
1514a1i 10 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( w  =  z  ->  ( ( w  e.  x  <->  w  e.  y )  <->  ( z  e.  x  <->  z  e.  y ) ) ) )
163, 11, 15cbvald 1948 . 2  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( A. w ( w  e.  x  <->  w  e.  y
)  <->  A. z ( z  e.  x  <->  z  e.  y ) ) )
17 axext3 2266 . 2  |-  ( A. w ( w  e.  x  <->  w  e.  y
)  ->  x  =  y )
1816, 17syl6bir 220 1  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  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   F/_wnfc 2406
This theorem is referenced by:  axext4dist  24157
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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