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Theorem ax8 1904
Description: Proof of ax-8 1900 from ax8v1 1902 and ax8v2 1903, proving sufficiency of the conjunction of the latter two weakened versions of ax8v 1901, which is itself a weakened version of ax-8 1900. (Contributed by BJ, 7-Dec-2020.)
Assertion
Ref Expression
ax8  |-  ( x  =  y  ->  (
x  e.  z  -> 
y  e.  z ) )

Proof of Theorem ax8
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 equviniv 1883 . 2  |-  ( x  =  y  ->  E. t
( x  =  t  /\  y  =  t ) )
2 equcomiv 1869 . . . . 5  |-  ( y  =  t  ->  t  =  y )
32adantl 472 . . . 4  |-  ( ( x  =  t  /\  y  =  t )  ->  t  =  y )
4 ax8v2 1903 . . . . 5  |-  ( x  =  t  ->  (
x  e.  z  -> 
t  e.  z ) )
54adantr 471 . . . 4  |-  ( ( x  =  t  /\  y  =  t )  ->  ( x  e.  z  ->  t  e.  z ) )
6 ax8v1 1902 . . . 4  |-  ( t  =  y  ->  (
t  e.  z  -> 
y  e.  z ) )
73, 5, 6sylsyld 58 . . 3  |-  ( ( x  =  t  /\  y  =  t )  ->  ( x  e.  z  ->  y  e.  z ) )
87exlimiv 1787 . 2  |-  ( E. t ( x  =  t  /\  y  =  t )  ->  (
x  e.  z  -> 
y  e.  z ) )
91, 8syl 17 1  |-  ( x  =  y  ->  (
x  e.  z  -> 
y  e.  z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375   E.wex 1674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675
This theorem is referenced by:  elequ1  1905  el  4602  axextdfeq  30494  ax8dfeq  30495  exnel  30499  bj-ax89  31322  bj-el  31457
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