Theorem supeq1i 7054

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 7052	. 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 1364  supcsup 7048

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  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-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-uni 3840  df-sup 7050

This theorem is referenced by:  infrenegsupex  9668  maxcom  11368  xrmax2sup  11419  xrmaxltsup  11423  xrmaxadd  11426  infxrnegsupex  11428  gcdcom  12140  gcdass  12182