Theorem appslamed ≪ | index | src | ≫ 

theorem appslamed (G: wff) (a b c: nat) {x: nat} (A: set x):
  $ G /\ x = a -> A @ b = c $ >
  $ G -> (\\ x, A) @ (a, b) = c $;
StepHypRefExpression
1 appslams 
(\\ x, A) @ (a, b) = (S[a / x] A) @ b
2 ax_6 
E. x x = a
3 nfv 
F/ x G
4 nfsbs1 
FS/ x S[a / x] A
5 nfnv 
FN/ x b
6 4, 5 nfapp 
FN/ x (S[a / x] A) @ b
7 nfnv 
FN/ x c
8 6, 7 nf_eq 
F/ x (S[a / x] A) @ b = c
9 sbsq 
x = a -> A == S[a / x] A
10 9 anwr 
G /\ x = a -> A == S[a / x] A
11 10 appeq1d 
G /\ x = a -> A @ b = (S[a / x] A) @ b
12 hyp h 
G /\ x = a -> A @ b = c
13 11, 12 eqtr3d 
G /\ x = a -> (S[a / x] A) @ b = c
14 13 exp 
G -> x = a -> (S[a / x] A) @ b = c
15 3, 8, 14 eexdh 
G -> E. x x = a -> (S[a / x] A) @ b = c
16 2, 15 mpi 
G -> (S[a / x] A) @ b = c
17 1, 16 syl5eq 
G -> (\\ x, A) @ (a, b) = 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), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)