Theorem ssabeled ≪ | index | src | ≫ 

theorem ssabeled (A: set) (G: wff) {x: nat} (a: nat) (p: wff x) (q: wff):
  $ G /\ x = a -> p -> q /\ x e. A $ >
  $ G -> a e. {x | p} -> q /\ a e. A $;
StepHypRefExpression
1 hyp h 
G /\ x = a -> p -> q /\ x e. A
2 anr 
G /\ x = a -> x = a
3 2 eleq1d 
G /\ x = a -> (x e. A <-> a e. A)
4 3 aneq2d 
G /\ x = a -> (q /\ x e. A <-> q /\ a e. A)
5 4 bi1d 
G /\ x = a -> q /\ x e. A -> q /\ a e. A
6 1, 5 syld 
G /\ x = a -> p -> q /\ a e. A
7 6 ssabed 
G -> a e. {x | p} -> q /\ a e. A

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)