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Theorem releq 4508
Description: Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
releq  |-  ( A  =  B  ->  ( Rel  A  <->  Rel  B ) )

Proof of Theorem releq
StepHypRef Expression
1 sseq1 3045 . 2  |-  ( A  =  B  ->  ( A  C_  ( _V  X.  _V )  <->  B  C_  ( _V 
X.  _V ) ) )
2 df-rel 4435 . 2  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
3 df-rel 4435 . 2  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
41, 2, 33bitr4g 221 1  |-  ( A  =  B  ->  ( Rel  A  <->  Rel  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   _Vcvv 2619    C_ wss 2997    X. cxp 4426   Rel wrel 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010  df-rel 4435
This theorem is referenced by:  releqi  4509  releqd  4510  dfrel2  4868  tposfn2  6013  ereq1  6279
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