Theorem gcdeqd ≪ | index | src | ≫ 

theorem gcdeqd (_G: wff) (_a1 _a2 _b1 _b2: nat):
  $ _G -> _a1 = _a2 $ >
  $ _G -> _b1 = _b2 $ >
  $ _G -> gcd _a1 _b1 = gcd _a2 _b2 $;
StepHypRefExpression
1 biidd 
_G -> (x || d <-> x || d)
2 eqidd 
_G -> x = x
3 hyp _ah 
_G -> _a1 = _a2
4 2, 3 dvdeqd 
_G -> (x || _a1 <-> x || _a2)
5 hyp _bh 
_G -> _b1 = _b2
6 2, 5 dvdeqd 
_G -> (x || _b1 <-> x || _b2)
7 4, 6 aneqd 
_G -> (x || _a1 /\ x || _b1 <-> x || _a2 /\ x || _b2)
8 1, 7 bieqd 
_G -> (x || d <-> x || _a1 /\ x || _b1 <-> (x || d <-> x || _a2 /\ x || _b2))
9 8 aleqd 
_G -> (A. x (x || d <-> x || _a1 /\ x || _b1) <-> A. x (x || d <-> x || _a2 /\ x || _b2))
10 9 abeqd 
_G -> {d | A. x (x || d <-> x || _a1 /\ x || _b1)} == {d | A. x (x || d <-> x || _a2 /\ x || _b2)}
11 10 theeqd 
_G -> the {d | A. x (x || d <-> x || _a1 /\ x || _b1)} = the {d | A. x (x || d <-> x || _a2 /\ x || _b2)}
12 11 conv gcd 
_G -> gcd _a1 _b1 = gcd _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)