Jelly, 14 bytes

Do2ÆPẠ æRÇÐf.ị 

Try it online!

Thanks to Erik the Outgolfer for the help fixing a bug. Thanks to Mr. Xcoder for the .ị trick.

Jelly, 14 bytes

Do2ÆPẠ æRÇÐf.ị 

Try it online!

æRÇÐf.ị Main link æR Prime range Ðf Filter the range with... Ç The helper link ị At index (decimal, returns [xs[floor], xs[ceil]], otherwise 0) . 0.5 Do2ÆPẠ Helper link D For each decimal o2 Replace 0s with 2s, an actual prime prime (could be 3, 5, or 7). ÆP Filter primes (1 if true, 0 if false) Ạ Check if all are true 

Thanks to Erik the Outgolfer for the help fixing a bug. Thanks to Mr. Xcoder for the .ị trick.

Jelly, 12 bytes

QÆPẠ æRÇÐf.ị 

Try it online!

Thanks to Erik the Outgolfer for the help fixing a bug. Thanks to Mr. Xcoder for the .ị trick.

Jelly, 14 bytes

Do2ÆPẠ æRÇÐf.ị 

Try it online!

Thanks to Erik the Outgolfer for the help fixing a bug. Thanks to Mr. Xcoder for the .ị trick.

Jelly, 21 bytes

QÆPẠ æRÇÐf çµḷ/,Ṫµ0¬? 

Try it online!

Thanks to Erik the Outgolfer for the help fixing a bug!

Jelly, 12 bytes

QÆPẠ æRÇÐf.ị 

Try it online!

Thanks to Erik the Outgolfer for the help fixing a bug. Thanks to Mr. Xcoder for the .ị trick.