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Theorem infeq1d 6614
Description: Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
Hypothesis
Ref Expression
infeq1d.1  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
infeq1d  |-  ( ph  -> inf ( B ,  A ,  R )  = inf ( C ,  A ,  R ) )

Proof of Theorem infeq1d
StepHypRef Expression
1 infeq1d.1 . 2  |-  ( ph  ->  B  =  C )
2 infeq1 6613 . 2  |-  ( B  =  C  -> inf ( B ,  A ,  R
)  = inf ( C ,  A ,  R
) )
31, 2syl 14 1  |-  ( ph  -> inf ( B ,  A ,  R )  = inf ( C ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285  infcinf 6585
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-uni 3628  df-sup 6586  df-inf 6587
This theorem is referenced by:  lcmval  10825  lcmass  10847
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