Theorem infeq1d 7342

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 7341	. 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 1402  infcinf 7313

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-uni 3931  df-sup 7314  df-inf 7315

This theorem is referenced by:  infssfzcldc  10647  infssfzledc  10648  zsupssdc  10651  xrbdtri  12020  nnmindc  12789  nnminle  12790  lcmval  12819  lcmass  12841  odzval  12998  ballotfilemi  13221  ballotfi  13260  nninfdclemcl  13317  nninfdclemp1  13319  nninfdc  13322  bdmetval  15524  bdxmet  15525  qtopbasss  15545  hovera  15671  hoverb  15672  hoverlt1  15673  hovergt0  15674  ivthdich  15677  repiecele0  16980  repiecege0  16981  repiecef  16982