Theorem leasteqd ≪ | index | src | ≫

theorem leasteqd (_G: wff) (_A1 _A2: set):
  $ _G -> _A1 == _A2 $ >
  $ _G -> least _A1 = least _A2 $;

Step	Hyp	Ref	Expression
1		hyp _Ah	_G -> _A1 == _A2
2	1	eleq2d	_G -> (x e. _A1 <-> x e. _A2)
3	1	eleq2d	_G -> (y e. _A1 <-> y e. _A2)
4	3	imeq1d	_G -> (y e. _A1 -> x <= y <-> y e. _A2 -> x <= y)
5	4	aleqd	_G -> (A. y (y e. _A1 -> x <= y) <-> A. y (y e. _A2 -> x <= y))
6	2, 5	aneqd	_G -> (x e. _A1 /\ A. y (y e. _A1 -> x <= y) <-> x e. _A2 /\ A. y (y e. _A2 -> x <= y))
7	6	abeqd	_G -> {x \| x e. _A1 /\ A. y (y e. _A1 -> x <= y)} == {x \| x e. _A2 /\ A. y (y e. _A2 -> x <= y)}
8	7	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)}
9	8	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)