Theorem dvdgcdlem ≪ | index | src | ≫ 

theorem dvdgcdlem (G: wff) (a b c d: nat) {x: nat}:
  $ G -> (x || d <-> x || a /\ x || b) $ >
  $ G -> (c || gcd a b <-> c || a /\ c || b) $;
StepHypRefExpression
1 hyp h 
G -> (x || d <-> x || a /\ x || b)
2 1 eqgcd 
G -> gcd a b = d
3 2 dvdeq2d 
G -> (c || gcd a b <-> c || d)
4 dvdeq1 
x = c -> (x || d <-> c || d)
5 dvdeq1 
x = c -> (x || a <-> c || a)
6 dvdeq1 
x = c -> (x || b <-> c || b)
7 5, 6 aneqd 
x = c -> (x || a /\ x || b <-> c || a /\ c || b)
8 4, 7 bieqd 
x = c -> (x || d <-> x || a /\ x || b <-> (c || d <-> c || a /\ c || b))
9 8 eale 
A. x (x || d <-> x || a /\ x || b) -> (c || d <-> c || a /\ c || b)
10 1 iald 
G -> A. x (x || d <-> x || a /\ x || b)
11 9, 10 syl 
G -> (c || d <-> c || a /\ c || b)
12 3, 11 bitrd 
G -> (c || gcd a b <-> c || a /\ c || b)

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), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)