Theorem supeq1i 6989

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 6987	. 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 1353   supcsup 6983

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-uni 3812  df-sup 6985

This theorem is referenced by:  infrenegsupex  9596  maxcom  11214  xrmax2sup  11264  xrmaxltsup  11268  xrmaxadd  11271  infxrnegsupex  11273  gcdcom  11976  gcdass  12018