Theorem infeq1d 7087

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 7086	. 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 1364  infcinf 7058

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-uni 3841  df-sup 7059  df-inf 7060

This theorem is referenced by:  zsupssdc  10345  xrbdtri  11458  nnmindc  12226  nnminle  12227  lcmval  12256  lcmass  12278  odzval  12435  nninfdclemcl  12690  nninfdclemp1  12692  nninfdc  12695  bdmetval  14820  bdxmet  14821  qtopbasss  14841  hovera  14967  hoverb  14968  hoverlt1  14969  hovergt0  14970  ivthdich  14973