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Theorem releq 4450
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 2994 . 2  |-  ( A  =  B  ->  ( A  C_  ( _V  X.  _V )  <->  B  C_  ( _V 
X.  _V ) ) )
2 df-rel 4380 . 2  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
3 df-rel 4380 . 2  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
41, 2, 33bitr4g 216 1  |-  ( A  =  B  ->  ( Rel  A  <->  Rel  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259   _Vcvv 2574    C_ wss 2945    X. cxp 4371   Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2952  df-ss 2959  df-rel 4380
This theorem is referenced by:  releqi  4451  releqd  4452  dfrel2  4799  tposfn2  5912  ereq1  6144
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