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Theorem ax12olem3 2007
Description: Lemma for nfeqf 2009 and dveeq1 2021. Convert ax12olem2 2006 into a more general form. (Contributed by Wolf Lammen, 29-Apr-2018.)
Hypothesis
Ref Expression
ax12olem3.1  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )
Assertion
Ref Expression
ax12olem3  |-  ( -. 
A. x  x  =  y  ->  F/ x  y  =  z )
Distinct variable group:    x, z

Proof of Theorem ax12olem3
StepHypRef Expression
1 exnal 1583 . 2  |-  ( E. x  -.  x  =  y  <->  -.  A. x  x  =  y )
2 nfnf1 1808 . . 3  |-  F/ x F/ x  y  =  z
3 ax12olem2 2006 . . . . 5  |-  ( -.  x  =  y  -> 
( E. x  y  =  z  ->  y  =  z ) )
4 ax12olem3.1 . . . . 5  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )
53, 4syld 42 . . . 4  |-  ( -.  x  =  y  -> 
( E. x  y  =  z  ->  A. x  y  =  z )
)
6 nf2 1889 . . . 4  |-  ( F/ x  y  =  z  <-> 
( E. x  y  =  z  ->  A. x  y  =  z )
)
75, 6sylibr 204 . . 3  |-  ( -.  x  =  y  ->  F/ x  y  =  z )
82, 7exlimi 1821 . 2  |-  ( E. x  -.  x  =  y  ->  F/ x  y  =  z )
91, 8sylbir 205 1  |-  ( -. 
A. x  x  =  y  ->  F/ x  y  =  z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549   E.wex 1550   F/wnf 1553
This theorem is referenced by:  nfeqf  2009  dveeq1  2021
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-6 1744  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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