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Theorem ax12olem3 1962
Description: Lemma for ax12o 1964. (Contributed by Wolf Lammen, 30-Jan-2018.)
Hypotheses
Ref Expression
ax12olem3.1  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )
ax12olem3.2  |-  ( -.  x  =  y  -> 
( y  =  w  ->  A. x  y  =  w ) )
Assertion
Ref Expression
ax12olem3  |-  ( -. 
A. x  x  =  y  ->  F/ x  y  =  z )
Distinct variable groups:    x, w, z    y, w

Proof of Theorem ax12olem3
StepHypRef Expression
1 exnal 1580 . 2  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
2 nfnf1 1798 . . 3  |-  F/ x F/ x  y  =  z
3 ax12olem3.2 . . . . . 6  |-  ( -.  x  =  y  -> 
( y  =  w  ->  A. x  y  =  w ) )
43ax12olem2 1961 . . . . 5  |-  ( -.  x  =  y  -> 
( E. x  y  =  z  ->  y  =  z ) )
5 ax12olem3.1 . . . . 5  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )
64, 5syld 42 . . . 4  |-  ( -.  x  =  y  -> 
( E. x  y  =  z  ->  A. x  y  =  z )
)
7 nf2 1878 . . . 4  |-  ( F/ x  y  =  z  <-> 
( E. x  y  =  z  ->  A. x  y  =  z )
)
86, 7sylibr 204 . . 3  |-  ( -.  x  =  y  ->  F/ x  y  =  z )
92, 8exlimi 1811 . 2  |-  ( E. x  -.  x  =  y  ->  F/ x  y  =  z )
101, 9sylbir 205 1  |-  ( -. 
A. x  x  =  y  ->  F/ x  y  =  z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1546   E.wex 1547   F/wnf 1550
This theorem is referenced by:  ax12o  1964  dvelimv  1975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
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