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Although "Don't let down (me)" sounds like the opposite to "Don't leave hanging(me)" the two phrases are synonymous Both phrases are about support or more precisely about losing needed support. If you ask another not to let you down you're asking that person not to take away their support. On the other hand if you ask a person not to leave you hanging, again you are asking them not to take away their support and leave you "hanging"
I need your help, please don't let down (me). I'm depending on your help, don't leave hanging (me).