Theorem cpleqd ≪ | index | src | ≫

theorem cpleqd (_G: wff) (_A1 _A2: set):
  $ _G -> _A1 == _A2 $ >
  $ _G -> Compl _A1 == Compl _A2 $;

Step	Hyp	Ref	Expression
1		eqidd	_G -> x = x
2		hyp _Ah	_G -> _A1 == _A2
3	1, 2	eleqd	_G -> (x e. _A1 <-> x e. _A2)
4	3	noteqd	_G -> (~x e. _A1 <-> ~x e. _A2)
5	4	abeqd	_G -> {x \| ~x e. _A1} == {x \| ~x e. _A2}
6	5	conv Compl	_G -> Compl _A1 == Compl _A2

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)