Theorem elsabed ≪ | index | src | ≫ 

theorem elsabed (G: wff) (a b: nat) (p: wff) {x: nat} (A: set x):
  $ G /\ x = a -> (b e. A <-> p) $ >
  $ G -> (a, b e. S\ x, A <-> p) $;
StepHypRefExpression
1 elsabs 
a, b e. S\ x, A <-> b e. 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 4 nfel2 
F/ x b e. S[a / x] A
6 nfv 
F/ x p
7 5, 6 nfbi 
F/ x b e. S[a / x] A <-> p
8 sbsq 
x = a -> A == S[a / x] A
9 8 anwr 
G /\ x = a -> A == S[a / x] A
10 9 eleq2d 
G /\ x = a -> (b e. A <-> b e. S[a / x] A)
11 hyp h 
G /\ x = a -> (b e. A <-> p)
12 10, 11 bitr3d 
G /\ x = a -> (b e. S[a / x] A <-> p)
13 12 exp 
G -> x = a -> (b e. S[a / x] A <-> p)
14 3, 7, 13 eexdh 
G -> E. x x = a -> (b e. S[a / x] A <-> p)
15 2, 14 mpi 
G -> (b e. S[a / x] A <-> p)
16 1, 15 syl5bb 
G -> (a, b e. S\ x, A <-> 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 (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)