Theorem elsabs ≪ | index | src | ≫ 

theorem elsabs (a b: nat) {x: nat} (A: set x):
  $ a, b e. S\ x, A <-> b e. S[a / x] A $;
StepHypRefExpression
1 sndpr 
snd (a, b) = b
2 sndeq 
p = a, b -> snd p = snd (a, b)
3 1, 2 syl6eq 
p = a, b -> snd p = b
4 fstpr 
fst (a, b) = a
5 fsteq 
p = a, b -> fst p = fst (a, b)
6 4, 5 syl6eq 
p = a, b -> fst p = a
7 6 sbseq1d 
p = a, b -> S[fst p / x] A == S[a / x] A
8 3, 7 eleqd 
p = a, b -> (snd p e. S[fst p / x] A <-> b e. S[a / x] A)
9 8 elabe 
a, b e. {p | snd p e. S[fst p / x] A} <-> b e. S[a / x] A
10 9 conv sab 
a, b e. S\ x, A <-> b e. S[a / x] A

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)