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Theorem infeq1d 6892
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 6891 . 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 1331  infcinf 6863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-uni 3732  df-sup 6864  df-inf 6865
This theorem is referenced by:  xrbdtri  11038  lcmval  11733  lcmass  11755  bdmetval  12658  bdxmet  12659  qtopbasss  12679
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