Theorem coprimeeqd ≪ | index | src | ≫

theorem coprimeeqd (_G: wff) (_a1 _a2 _b1 _b2: nat):
  $ _G -> _a1 = _a2 $ >
  $ _G -> _b1 = _b2 $ >
  $ _G -> (coprime _a1 _b1 <-> coprime _a2 _b2) $;


_G -> _a1 = _a2
_G -> _b1 = _b2
_G -> gcd _a1 _b1 = gcd _a2 _b2
_G -> 1 = 1
_G -> (gcd _a1 _b1 = 1 <-> gcd _a2 _b2 = 1)
_G -> (coprime _a1 _b1 <-> coprime _a2 _b2)

Step	Hyp	Ref	Expression
1		hyp _ah	_G -> _a1 = _a2
2		hyp _bh	_G -> _b1 = _b2
3	1, 2	gcdeqd	_G -> gcd _a1 _b1 = gcd _a2 _b2
4		eqidd	_G -> 1 = 1
5	3, 4	eqeqd	_G -> (gcd _a1 _b1 = 1 <-> gcd _a2 _b2 = 1)
6	5	conv coprime	_G -> (coprime _a1 _b1 <-> coprime _a2 _b2)

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (muleq)