Theorem infeq1d 7210

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 7209	. 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 1397  infcinf 7181

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-uni 3894  df-sup 7182  df-inf 7183

This theorem is referenced by:  zsupssdc  10497  xrbdtri  11836  nnmindc  12604  nnminle  12605  lcmval  12634  lcmass  12656  odzval  12813  nninfdclemcl  13068  nninfdclemp1  13070  nninfdc  13073  bdmetval  15223  bdxmet  15224  qtopbasss  15244  hovera  15370  hoverb  15371  hoverlt1  15372  hovergt0  15373  ivthdich  15376