Theorem sappslamed ≪ | index | src | ≫ 

theorem sappslamed (B: set) (G: wff) (a: nat) {x: nat} (A: set x):
  $ G /\ x = a -> A == B $ >
  $ G -> (\\ x, A) @@ a == B $;
StepHypRefExpression
1 sappslams 
(\\ x, A) @@ a == S[a / x] A
2 ax_6 
E. x x = a
3 nfv 
F/ x G
4 nfsbs1 
FS/ x S[a / x] A
5 nfsv 
FS/ x B
6 4, 5 nfeqs 
F/ x S[a / x] A == B
7 sbsq 
x = a -> A == S[a / x] A
8 7 anwr 
G /\ x = a -> A == S[a / x] A
9 hyp h 
G /\ x = a -> A == B
10 8, 9 eqstr3d 
G /\ x = a -> S[a / x] A == B
11 10 exp 
G -> x = a -> S[a / x] A == B
12 3, 6, 11 eexdh 
G -> E. x x = a -> S[a / x] A == B
13 2, 12 mpi 
G -> S[a / x] A == B
14 1, 13 syl5eqs 
G -> (\\ x, A) @@ a == 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, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)