Theorem inseqd ≪ | index | src | ≫

theorem inseqd (_G: wff) (_a1 _a2 _b1 _b2: nat):
  $ _G -> _a1 = _a2 $ >
  $ _G -> _b1 = _b2 $ >
  $ _G -> _a1 ; _b1 = _a2 ; _b2 $;

Step	Hyp	Ref	Expression
1		eqidd	_G -> x = x
2		hyp _ah	_G -> _a1 = _a2
3	1, 2	eqeqd	_G -> (x = _a1 <-> x = _a2)
4		hyp _bh	_G -> _b1 = _b2
5	4	nseqd	_G -> _b1 == _b2
6	1, 5	eleqd	_G -> (x e. _b1 <-> x e. _b2)
7	3, 6	oreqd	_G -> (x = _a1 \/ x e. _b1 <-> x = _a2 \/ x e. _b2)
8	7	abeqd	_G -> {x \| x = _a1 \/ x e. _b1} == {x \| x = _a2 \/ x e. _b2}
9	8	lowereqd	_G -> lower {x \| x = _a1 \/ x e. _b1} = lower {x \| x = _a2 \/ x e. _b2}
10	9	conv ins	_G -> _a1 ; _b1 = _a2 ; _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), axs_the (theid, the0), axs_peano (peano2, addeq, muleq)