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Theorem bj-elequ12 31323
Description: An identity law for the non-logical predicate, which combines elequ1 1905 and elequ2 1912. For the analogous theorems for class terms, see eleq1 2528, eleq2 2529 and eleq12 2530. (TODO: move to main part.) (Contributed by BJ, 29-Sep-2019.)
Assertion
Ref Expression
bj-elequ12  |-  ( ( x  =  y  /\  z  =  t )  ->  ( x  e.  z  <-> 
y  e.  t ) )

Proof of Theorem bj-elequ12
StepHypRef Expression
1 elequ1 1905 . 2  |-  ( x  =  y  ->  (
x  e.  z  <->  y  e.  z ) )
2 elequ2 1912 . 2  |-  ( z  =  t  ->  (
y  e.  z  <->  y  e.  t ) )
31, 2sylan9bb 711 1  |-  ( ( x  =  y  /\  z  =  t )  ->  ( x  e.  z  <-> 
y  e.  t ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375
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  ax-9 1907
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675
This theorem is referenced by:  bj-ru0  31583
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