Theorem splitpr1 ≪ | index | src | ≫ 

theorem splitpr1 (G: wff) (a b: nat) (p: wff) (x y: nat):
  $ G -> a = x, y -> p $ >
  $ G -> b = x -> a = b, y -> p $;
StepHypRefExpression
1 hyp h 
G -> a = x, y -> p
2 anll 
G /\ b = x /\ a = b, y -> G
3 anr 
G /\ b = x /\ a = b, y -> a = b, y
4 anlr 
G /\ b = x /\ a = b, y -> b = x
5 4 preq1d 
G /\ b = x /\ a = b, y -> b, y = x, y
6 3, 5 eqtrd 
G /\ b = x /\ a = b, y -> a = x, y
7 1, 2, 6 sylc 
G /\ b = x /\ a = b, y -> p
8 7 exp 
G /\ b = x -> a = b, y -> p
9 8 exp 
G -> b = x -> a = b, y -> p

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 (peano2, addeq, muleq)