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

Step	Hyp	Ref	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 ) )
4	1, 2, 3	3bitr4g 292	1 |- ( A = B -> ( Rel A <-> Rel B ) )

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