Theorem ineqd ≪ | index | src | ≫

theorem ineqd (_G: wff) (_A1 _A2 _B1 _B2: set):
  $ _G -> _A1 == _A2 $ >
  $ _G -> _B1 == _B2 $ >
  $ _G -> _A1 i^i _B1 == _A2 i^i _B2 $;

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		hyp _Bh	_G -> _B1 == _B2
5	1, 4	eleqd	_G -> (x e. _B1 <-> x e. _B2)
6	3, 5	aneqd	_G -> (x e. _A1 /\ x e. _B1 <-> x e. _A2 /\ x e. _B2)
7	6	abeqd	_G -> {x \| x e. _A1 /\ x e. _B1} == {x \| x e. _A2 /\ x e. _B2}
8	7	conv Inter	_G -> _A1 i^i _B1 == _A2 i^i _B2

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)