Theorem releqi 4747

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 4746	. 2 |- ( A = B -> ( Rel A <-> Rel B ) )
3	1, 2	ax-mp 5	1 |- ( Rel A <-> Rel B )

Syntax hints:   <-> wb 105   = wceq 1364   Rel wrel 4669

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170  df-rel 4671

This theorem is referenced by:  reliun  4785  reluni  4787  relint  4788  reldmmpo  6038  tfrlem6  6383  subrgdvds  13867  rrgmex  13893  lssmex  13987  2idlmex  14133  psmetrel  14642  metrel  14662  xmetrel  14663  xmetf  14670  mopnrel  14761