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Theorem infeq1d 6867
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 6866 . 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 1316  infcinf 6838
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-uni 3707  df-sup 6839  df-inf 6840
This theorem is referenced by:  xrbdtri  11013  lcmval  11671  lcmass  11693  bdmetval  12596  bdxmet  12597  qtopbasss  12617
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