Theorem releqi 4758

Description: Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)

Hypothesis

Ref	Expression
releqi.1	|- A = B

Assertion

Ref	Expression
releqi	|- ( Rel A <-> Rel B )

Proof of Theorem releqi

Step	Hyp	Ref	Expression
1		releqi.1	. 2 |- A = B
2		releq 4757	. 2 |- ( A = B -> ( Rel A <-> Rel B ) )
3	1, 2	ax-mp 5	1 |- ( Rel A <-> Rel B )

Syntax hints:   <-> wb 105   = wceq 1373   Rel wrel 4680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179  df-rel 4682

This theorem is referenced by:  reliun  4796  reluni  4798  relint  4799  reldmmpo  6057  tfrlem6  6402  subrgdvds  13997  rrgmex  14023  lssmex  14117  2idlmex  14263  psmetrel  14794  metrel  14814  xmetrel  14815  xmetf  14822  mopnrel  14913