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Theorem axextdist 30495
Description: ax-ext 2442 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 2163 . . . 4  |-  F/ z  -.  A. z  z  =  x
2 nfnae 2163 . . . 4  |-  F/ z  -.  A. z  z  =  y
31, 2nfan 2022 . . 3  |-  F/ z ( -.  A. z 
z  =  x  /\  -.  A. z  z  =  y )
4 nfcvf 2626 . . . . . 6  |-  ( -. 
A. z  z  =  x  ->  F/_ z x )
54adantr 471 . . . . 5  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/_ z x )
65nfcrd 2609 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  w  e.  x )
7 nfcvf 2626 . . . . . 6  |-  ( -. 
A. z  z  =  y  ->  F/_ z y )
87adantl 472 . . . . 5  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/_ z y )
98nfcrd 2609 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  w  e.  y )
106, 9nfbid 2027 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z
( w  e.  x  <->  w  e.  y ) )
11 elequ1 1905 . . . . 5  |-  ( w  =  z  ->  (
w  e.  x  <->  z  e.  x ) )
12 elequ1 1905 . . . . 5  |-  ( w  =  z  ->  (
w  e.  y  <->  z  e.  y ) )
1311, 12bibi12d 327 . . . 4  |-  ( w  =  z  ->  (
( w  e.  x  <->  w  e.  y )  <->  ( z  e.  x  <->  z  e.  y ) ) )
1413a1i 11 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( w  =  z  ->  ( ( w  e.  x  <->  w  e.  y )  <->  ( z  e.  x  <->  z  e.  y ) ) ) )
153, 10, 14cbvald 2129 . 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 ) ) )
16 axext3 2444 . 2  |-  ( A. w ( w  e.  x  <->  w  e.  y
)  ->  x  =  y )
1715, 16syl6bir 237 1  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1453   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:  axext4dist  30496
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