Theorem releq 5073

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 3510	. 2 |- ( A = B -> ( A C_ ( _V X. _V ) <-> B C_ ( _V X. _V ) ) )
2		df-rel 4995	. 2 |- ( Rel A <-> A C_ ( _V X. _V ) )
3		df-rel 4995	. 2 |- ( Rel B <-> B C_ ( _V X. _V ) )
4	1, 2, 3	3bitr4g 288	1 |- ( A = B -> ( Rel A <-> Rel B ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1398   _Vcvv 3106   C_ wss 3461   X. cxp 4986   Rel wrel 4993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-in 3468  df-ss 3475  df-rel 4995

This theorem is referenced by:  releqi  5074  releqd  5075  dfrel2  5441  tposfn2  6969  ereq1  7310  isps  16034  isdir  16064  fpwrelmapffslem  27789  frrlem6  29639  prtlem12  30851  bnj1321  34503