Theorem supeq1i 6763

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 6761	. 2 |- ( B = C -> sup ( B , A , R ) = sup ( C , A , R ) )
3	1, 2	ax-mp 7	1 |- sup ( B , A , R ) = sup ( C , A , R )

Syntax hints:   = wceq 1296   supcsup 6757

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-uni 3676  df-sup 6759

This theorem is referenced by:  infrenegsupex  9181  maxcom  10751  xrmax2sup  10797  xrmaxltsup  10801  xrmaxadd  10804  infxrnegsupex  10806  gcdcom  11392  gcdass  11431