Theorem splitb0 ≪ | index | src | ≫ 

theorem splitb0 (G: wff) (a b c: nat) (p: wff):
  $ G -> a = b0 c -> p $ >
  $ G -> b = c -> a = b0 b -> p $;
StepHypRefExpression
1 hyp h 
G -> a = b0 c -> p
2 anll 
G /\ b = c /\ a = b0 b -> G
3 anr 
G /\ b = c /\ a = b0 b -> a = b0 b
4 anlr 
G /\ b = c /\ a = b0 b -> b = c
5 4 b0eqd 
G /\ b = c /\ a = b0 b -> b0 b = b0 c
6 3, 5 eqtrd 
G /\ b = c /\ a = b0 b -> a = b0 c
7 1, 2, 6 sylc 
G /\ b = c /\ a = b0 b -> p
8 7 exp 
G /\ b = c -> a = b0 b -> p
9 8 exp 
G -> b = c -> a = b0 b -> p

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7), axs_peano (addeq)