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Theorem bj-ax89 31277
Description: A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 1892 and ax-9 1899. Indeed, it is implied over propositional calculus by the conjunction of ax-8 1892 and ax-9 1899, as proved here. In the other direction, one can prove ax-8 1892 (respectively ax-9 1899) from bj-ax89 31277 by using mpan2 682 ( respectively mpan 681) and equid 1858. (TODO: move to main part.) (Contributed by BJ, 3-Oct-2019.)
Assertion
Ref Expression
bj-ax89  |-  ( ( x  =  y  /\  z  =  t )  ->  ( x  e.  z  ->  y  e.  t ) )

Proof of Theorem bj-ax89
StepHypRef Expression
1 ax8 1896 . 2  |-  ( x  =  y  ->  (
x  e.  z  -> 
y  e.  z ) )
2 ax9 1903 . 2  |-  ( z  =  t  ->  (
y  e.  z  -> 
y  e.  t ) )
31, 2sylan9 667 1  |-  ( ( x  =  y  /\  z  =  t )  ->  ( x  e.  z  ->  y  e.  t ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-8 1892  ax-9 1899
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1667
This theorem is referenced by: (None)
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