Short answer
NO
POST variables are not 'safe' on their own.
Long Answer:
There are many issues around authenticating POSTed variables on your PHP script. The most well known is Cross Site Request Forgeries whereby the value is sent to your page but you have no idea where the value came from, or who sent it.
To counter this you need to set up single use unique keys to send and receive, typically using PHP SESSIONS or similar concepts that can not be touched by the end user.
The second issue is that you should never ever ever trust the contents of a POSTed variable. you need to fully escape the variable as well as ensuring that the variable is given absolutely minimum access to your code, so if it is a compromised value, it is hopefully cleaned and/or it can not harm your script.
Your Code:
Looking at your question in detail you suffer from CSRF (point 1, above) in that your PHP code has no idea if the submitted button came from the page your button exists on and therefore the PHP has no idea if the browser should see the contents of "Table A". (If everyone can see it, why hide it in the first place?)
There is also the factor your code could be used by a nefarious party to send many, many POSTed submits to the script, over a short space of time (100,000 over 2 seconds) causing a DOS attack, again due to there being no validation of where the POST came from and its implied authenticity.
Further points: You need to clarify what you deem as "safe", it is a very relative term.