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Theorem infeq1d 6707
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 6706 . 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 1289  infcinf 6678
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-uni 3654  df-sup 6679  df-inf 6680
This theorem is referenced by:  lcmval  11323  lcmass  11345
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