Theorem releqi 4860

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 4859	. 2 |- ( A = B -> ( Rel A <-> Rel B ) )
3	1, 2	ax-mp 5	1 |- ( Rel A <-> Rel B )

This proof depends on syntax axioms:   <-> wb 105   = wceq 1402   Rel wrel 4781

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233  df-rel 4783

This theorem is used by:  reliun  4900  reluni  4902  relint  4903  reldmmpo  6203  tfrlem6  6592  ringidval  14371  opprringb  14494  reldvdsr  14506  subrgdvds  14651  rrgmex  14677  lssmex  14800  2idlmex  14946  asclfval  15129  psmetrel  15565  metrel  15585  xmetrel  15586  xmetf  15593  mopnrel  15684