Theorem releq 4440

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

Step	Hyp	Ref	Expression
1		sseq1 3020	. 2 |- ( A = B -> ( A C_ ( _V X. _V ) <-> B C_ ( _V X. _V ) ) )
2		df-rel 4370	. 2 |- ( Rel A <-> A C_ ( _V X. _V ) )
3		df-rel 4370	. 2 |- ( Rel B <-> B C_ ( _V X. _V ) )
4	1, 2, 3	3bitr4g 221	1 |- ( A = B -> ( Rel A <-> Rel B ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   _Vcvv 2601   C_ wss 2973   X. cxp 4361   Rel wrel 4368

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 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986  df-rel 4370

This theorem is referenced by:  releqi  4441  releqd  4442  dfrel2  4791  tposfn2  5904  ereq1  6136