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Theorem infeq1 6650
Description: Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
Assertion
Ref Expression
infeq1  |-  ( B  =  C  -> inf ( B ,  A ,  R
)  = inf ( C ,  A ,  R
) )

Proof of Theorem infeq1
StepHypRef Expression
1 supeq1 6625 . 2  |-  ( B  =  C  ->  sup ( B ,  A ,  `' R )  =  sup ( C ,  A ,  `' R ) )
2 df-inf 6624 . 2  |- inf ( B ,  A ,  R
)  =  sup ( B ,  A ,  `' R )
3 df-inf 6624 . 2  |- inf ( C ,  A ,  R
)  =  sup ( C ,  A ,  `' R )
41, 2, 33eqtr4g 2142 1  |-  ( B  =  C  -> inf ( B ,  A ,  R
)  = inf ( C ,  A ,  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287   `'ccnv 4410   supcsup 6621  infcinf 6622
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-uni 3637  df-sup 6623  df-inf 6624
This theorem is referenced by:  infeq1d  6651  infeq1i  6652
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