Theorem leasteqd ≪ | index | src | ≫

theorem leasteqd (_G: wff) (_A1 _A2: set):
  $ _G -> _A1 == _A2 $ >
  $ _G -> least _A1 = least _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		eqidd	_G -> y = y
5	4, 2	eleqd	_G -> (y e. _A1 <-> y e. _A2)
6		biidd	_G -> (x <= y <-> x <= y)
7	5, 6	imeqd	_G -> (y e. _A1 -> x <= y <-> y e. _A2 -> x <= y)
8	7	aleqd	_G -> (A. y (y e. _A1 -> x <= y) <-> A. y (y e. _A2 -> x <= y))
9	3, 8	aneqd	_G -> (x e. _A1 /\ A. y (y e. _A1 -> x <= y) <-> x e. _A2 /\ A. y (y e. _A2 -> x <= y))
10	9	abeqd	_G -> {x \| x e. _A1 /\ A. y (y e. _A1 -> x <= y)} == {x \| x e. _A2 /\ A. y (y e. _A2 -> x <= y)}
11	10	theeqd	_G -> the {x \| x e. _A1 /\ A. y (y e. _A1 -> x <= y)} = the {x \| x e. _A2 /\ A. y (y e. _A2 -> x <= y)}
12	11	conv least	_G -> least _A1 = least _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), axs_the (theid, the0)