Theorem supeq1i 7116

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 7114	. 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 7110

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-uni 3865  df-sup 7112

This theorem is referenced by:  infrenegsupex  9750  maxcom  11629  xrmax2sup  11680  xrmaxltsup  11684  xrmaxadd  11687  infxrnegsupex  11689  gcdcom  12409  gcdass  12451