Theorem infeq1d 7114

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 7113	. 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 1373  infcinf 7085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-uni 3851  df-sup 7086  df-inf 7087

This theorem is referenced by:  zsupssdc  10381  xrbdtri  11587  nnmindc  12355  nnminle  12356  lcmval  12385  lcmass  12407  odzval  12564  nninfdclemcl  12819  nninfdclemp1  12821  nninfdc  12824  bdmetval  14972  bdxmet  14973  qtopbasss  14993  hovera  15119  hoverb  15120  hoverlt1  15121  hovergt0  15122  ivthdich  15125