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Theorem axextdist 25427
Description: ax-ext 2417 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 2044 . . . 4  |-  F/ z  -.  A. z  z  =  x
2 nfnae 2044 . . . 4  |-  F/ z  -.  A. z  z  =  y
31, 2nfan 1846 . . 3  |-  F/ z ( -.  A. z 
z  =  x  /\  -.  A. z  z  =  y )
4 nfcvf 2594 . . . . . 6  |-  ( -. 
A. z  z  =  x  ->  F/_ z x )
54adantr 452 . . . . 5  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/_ z x )
65nfcrd 2585 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  w  e.  x )
7 nfcvf 2594 . . . . . 6  |-  ( -. 
A. z  z  =  y  ->  F/_ z y )
87adantl 453 . . . . 5  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/_ z y )
98nfcrd 2585 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  w  e.  y )
106, 9nfbid 1854 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z
( w  e.  x  <->  w  e.  y ) )
11 elequ1 1728 . . . . 5  |-  ( w  =  z  ->  (
w  e.  x  <->  z  e.  x ) )
12 elequ1 1728 . . . . 5  |-  ( w  =  z  ->  (
w  e.  y  <->  z  e.  y ) )
1311, 12bibi12d 313 . . . 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 1986 . 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 2419 . 2  |-  ( A. w ( w  e.  x  <->  w  e.  y
)  ->  x  =  y )
1715, 16syl6bir 221 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 177    /\ wa 359   A.wal 1549   F/_wnfc 2559
This theorem is referenced by:  axext4dist  25428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-cleq 2429  df-clel 2432  df-nfc 2561
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