Theorem supeq1i 7090

Description: Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypothesis

Ref	Expression
supeq1i.1	|- B = C

Assertion

Ref	Expression
supeq1i	|- sup ( B , A , R ) = sup ( C , A , R )

Proof of Theorem supeq1i

Step	Hyp	Ref	Expression
1		supeq1i.1	. 2 |- B = C
2		supeq1 7088	. 2 |- ( B = C -> sup ( B , A , R ) = sup ( C , A , R ) )
3	1, 2	ax-mp 5	1 |- sup ( B , A , R ) = sup ( C , A , R )

Syntax hints:   = wceq 1373   supcsup 7084

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-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  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-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-uni 3851  df-sup 7086

This theorem is referenced by:  infrenegsupex  9715  maxcom  11514  xrmax2sup  11565  xrmaxltsup  11569  xrmaxadd  11572  infxrnegsupex  11574  gcdcom  12294  gcdass  12336