Theorem sbneq2d ≪ | index | src | ≫ 

theorem sbneq2d {x: nat} (G: wff) (a b c: nat x):
  $ G -> b = c $ >
  $ G -> N[a / x] b = N[a / x] c $;
StepHypRefExpression
1 anr 
G /\ y = z -> y = z
2 hyp h 
G -> b = c
3 2 anwl 
G /\ y = z -> b = c
4 1, 3 eqeqd 
G /\ y = z -> (y = b <-> z = c)
5 4 sbeq2d 
G /\ y = z -> ([a / x] y = b <-> [a / x] z = c)
6 5 cbvabd 
G -> {y | [a / x] y = b} == {z | [a / x] z = c}
7 6 theeqd 
G -> the {y | [a / x] y = b} = the {z | [a / x] z = c}
8 7 conv sbn 
G -> N[a / x] b = N[a / x] c

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)