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Theorem releq 4917
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 3453 . 2  |-  ( A  =  B  ->  ( A  C_  ( _V  X.  _V )  <->  B  C_  ( _V 
X.  _V ) ) )
2 df-rel 4841 . 2  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
3 df-rel 4841 . 2  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
41, 2, 33bitr4g 292 1  |-  ( A  =  B  ->  ( Rel  A  <->  Rel  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    = wceq 1444   _Vcvv 3045    C_ wss 3404    X. cxp 4832   Rel wrel 4839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-in 3411  df-ss 3418  df-rel 4841
This theorem is referenced by:  releqi  4918  releqd  4919  dfrel2  5286  tposfn2  6995  ereq1  7370  isps  16448  isdir  16478  fpwrelmapffslem  28317  bnj1321  29836  frrlem6  30523  prtlem12  32439  relintabex  36187  clrellem  36229  clcnvlem  36230
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