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Theorem releq 4962
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 3371 . 2  |-  ( A  =  B  ->  ( A  C_  ( _V  X.  _V )  <->  B  C_  ( _V 
X.  _V ) ) )
2 df-rel 4888 . 2  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
3 df-rel 4888 . 2  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
41, 2, 33bitr4g 281 1  |-  ( A  =  B  ->  ( Rel  A  <->  Rel  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653   _Vcvv 2958    C_ wss 3322    X. cxp 4879   Rel wrel 4886
This theorem is referenced by:  releqi  4963  releqd  4964  dfrel2  5324  tposfn2  6504  ereq1  6915  isps  14639  isdir  14682  relexprel  25139  frrlem6  25596  prtlem12  26730  bnj1321  29470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-in 3329  df-ss 3336  df-rel 4888
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