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Theorem ax12 2019
Description: Derive ax-12 1950 from ax12v 1951 via ax12o 2010. This shows that the weakening in ax12v 1951 is still sufficient for a complete system. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.)
Assertion
Ref Expression
ax12  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )

Proof of Theorem ax12
StepHypRef Expression
1 sp 1763 . . . 4  |-  ( A. x  x  =  y  ->  x  =  y )
21con3i 129 . . 3  |-  ( -.  x  =  y  ->  -.  A. x  x  =  y )
3 sp 1763 . . . 4  |-  ( A. x  x  =  z  ->  x  =  z )
43con3i 129 . . 3  |-  ( -.  x  =  z  ->  -.  A. x  x  =  z )
5 ax12o 2010 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
62, 4, 5syl2im 36 . 2  |-  ( -.  x  =  y  -> 
( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
7 ax12b 1701 . 2  |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)  <->  ( -.  x  =  y  ->  ( -.  x  =  z  -> 
( y  =  z  ->  A. x  y  =  z ) ) ) )
86, 7mpbir 201 1  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549
This theorem is referenced by:  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-7 1749  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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