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Theorem bj-ax89 34639
Description: A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 1825 and ax-9 1827. Indeed, it is implied over propositional calculus by the conjunction of ax-8 1825 and ax-9 1827, as proved here. In the other direction, one can prove ax-8 1825 (respectively ax-9 1827) from bj-ax89 34639 by using mpan2 669 ( respectively mpan 668) and equid 1796. (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 ax-8 1825 . 2  |-  ( x  =  y  ->  (
x  e.  z  -> 
y  e.  z ) )
2 ax-9 1827 . 2  |-  ( z  =  t  ->  (
y  e.  z  -> 
y  e.  t ) )
31, 2sylan9 655 1  |-  ( ( x  =  y  /\  z  =  t )  ->  ( x  e.  z  ->  y  e.  t ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-8 1825  ax-9 1827
This theorem depends on definitions:  df-bi 185  df-an 369
This theorem is referenced by: (None)
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