Theorem releq 5201

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 3626	. 2 |- ( A = B -> ( A C_ ( _V X. _V ) <-> B C_ ( _V X. _V ) ) )
2		df-rel 5121	. 2 |- ( Rel A <-> A C_ ( _V X. _V ) )
3		df-rel 5121	. 2 |- ( Rel B <-> B C_ ( _V X. _V ) )
4	1, 2, 3	3bitr4g 303	1 |- ( A = B -> ( Rel A <-> Rel B ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   _Vcvv 3200   C_ wss 3574   X. cxp 5112   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-rel 5121

This theorem is referenced by:  releqi  5202  releqd  5203  dfrel2  5583  tposfn2  7374  ereq1  7749  isps  17202  isdir  17232  fpwrelmapffslem  29507  bnj1321  31095  frrlem6  31789  prtlem12  34152  relintabex  37887  clrellem  37929  clcnvlem  37930