logic – Demonstrate $ ( exists x) (A a (( exists x) (A wedge (B vee C)) equiv B vee ( exists x) (A wedge C)) $, if $ x $ is not free at $ B $.

Try out $ ( exists x) (A a (( exists x) (A wedge (B vee C)) equiv B vee ( exists x) (A wedge C)) $, Yes $ x $ It is not free in $ B $.

I have tried several different ways and I can not see a good way forward. I feel that my best bet is to start from:

$$ A[x:=z] vdash (( exists x) (A wedge (B vee C)) equiv. B vee ( exists x) (A wedge C))[x:=z]$$

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Each projective module $ mathbb {Z} / p ^ n mathbb {Z} $ – is a free module $ mathbb {Z} / p ^ n mathbb {Z} $ – if p is a prime number

If p is a prime number, how to test each project? $ mathbb {Z} / p ^ n mathbb {Z} $-module is a free $ mathbb {Z} / p ^ n mathbb {Z} $-module?