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Theorem ax13 2103
Description: Derive ax-13 2054 from ax13v 2055 via axc9 2102. This shows that the weakening in ax13v 2055 is still sufficient for a complete system. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.)
Assertion
Ref Expression
ax13  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )

Proof of Theorem ax13
StepHypRef Expression
1 sp 1911 . . . 4  |-  ( A. x  x  =  y  ->  x  =  y )
21con3i 141 . . 3  |-  ( -.  x  =  y  ->  -.  A. x  x  =  y )
3 sp 1911 . . . 4  |-  ( A. x  x  =  z  ->  x  =  z )
43con3i 141 . . 3  |-  ( -.  x  =  z  ->  -.  A. x  x  =  z )
5 axc9 2102 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
62, 4, 5syl2im 40 . 2  |-  ( -.  x  =  y  -> 
( -.  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z )
) )
7 ax13b 1856 . 2  |-  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)  <->  ( -.  x  =  y  ->  ( -.  x  =  z  -> 
( y  =  z  ->  A. x  y  =  z ) ) ) )
86, 7mpbir 213 1  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-12 1906  ax-13 2054
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1661  df-nf 1665
This theorem is referenced by:  equvini  2143  sbequi  2170
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