Theorem fineqd ≪ | index | src | ≫ 

theorem fineqd (_G: wff) (_A1 _A2: set):
  $ _G -> _A1 == _A2 $ >
  $ _G -> (finite _A1 <-> finite _A2) $;
StepHypRefExpression
1 eqidd 
_G -> x = x
2 hyp _Ah 
_G -> _A1 == _A2
3 1, 2 eleqd 
_G -> (x e. _A1 <-> x e. _A2)
4 biidd 
_G -> (x < n <-> x < n)
5 3, 4 imeqd 
_G -> (x e. _A1 -> x < n <-> x e. _A2 -> x < n)
6 5 aleqd 
_G -> (A. x (x e. _A1 -> x < n) <-> A. x (x e. _A2 -> x < n))
7 6 exeqd 
_G -> (E. n A. x (x e. _A1 -> x < n) <-> E. n A. x (x e. _A2 -> x < n))
8 7 conv finite 
_G -> (finite _A1 <-> finite _A2)

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_12), axs_set (ax_8)