Theorem elxabed ≪ | index | src | ≫ 

theorem elxabed (A: set) (G: wff) (a b: nat) (p: wff) {x: nat} (B: set x):
  $ G /\ x = a -> (b e. B <-> p) $ >
  $ G -> (a, b e. X\ x e. A, B <-> a e. A /\ p) $;
StepHypRefExpression
1 elxabs 
a, b e. X\ x e. A, B <-> a e. A /\ b e. S[a / x] B
2 ax_6 
E. x x = a
3 nfv 
F/ x G
4 nfv 
F/ x a e. A
5 nfsbs1 
FS/ x S[a / x] B
6 5 nfel2 
F/ x b e. S[a / x] B
7 4, 6 nfan 
F/ x a e. A /\ b e. S[a / x] B
8 nfv 
F/ x a e. A /\ p
9 7, 8 nfbi 
F/ x a e. A /\ b e. S[a / x] B <-> a e. A /\ p
10 sbsq 
x = a -> B == S[a / x] B
11 10 anwr 
G /\ x = a -> B == S[a / x] B
12 11 eleq2d 
G /\ x = a -> (b e. B <-> b e. S[a / x] B)
13 hyp h 
G /\ x = a -> (b e. B <-> p)
14 12, 13 bitr3d 
G /\ x = a -> (b e. S[a / x] B <-> p)
15 14 aneq2d 
G /\ x = a -> (a e. A /\ b e. S[a / x] B <-> a e. A /\ p)
16 15 exp 
G -> x = a -> (a e. A /\ b e. S[a / x] B <-> a e. A /\ p)
17 3, 9, 16 eexdh 
G -> E. x x = a -> (a e. A /\ b e. S[a / x] B <-> a e. A /\ p)
18 2, 17 mpi 
G -> (a e. A /\ b e. S[a / x] B <-> a e. A /\ p)
19 1, 18 syl5bb 
G -> (a, b e. X\ x e. A, B <-> a e. 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)