Theorem infeq1d 6907

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

Step	Hyp	Ref	Expression
1		infeq1d.1	. 2 |- ( ph -> B = C )
2		infeq1 6906	. 2 |- ( B = C -> inf ( B , A , R ) = inf ( C , A , R ) )
3	1, 2	syl 14	1 |- ( ph -> inf ( B , A , R ) = inf ( C , A , R ) )

Syntax hints:   -> wi 4   = wceq 1332  infcinf 6878

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-uni 3745  df-sup 6879  df-inf 6880

This theorem is referenced by:  xrbdtri  11077  lcmval  11780  lcmass  11802  bdmetval  12708  bdxmet  12709  qtopbasss  12729