Theorem lteqd ≪ | index | src | ≫

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


_G -> _a1 = _a2
_G -> suc _a1 = suc _a2
_G -> _b1 = _b2
_G -> (suc _a1 <= _b1 <-> suc _a2 <= _b2)
_G -> (_a1 < _b1 <-> _a2 < _b2)

Step	Hyp	Ref	Expression
1		hyp _ah	_G -> _a1 = _a2
2	1	suceqd	_G -> suc _a1 = suc _a2
3		hyp _bh	_G -> _b1 = _b2
4	2, 3	leeqd	_G -> (suc _a1 <= _b1 <-> suc _a2 <= _b2)
5	4	conv lt	_G -> (_a1 < _b1 <-> _a2 < _b2)

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7), axs_peano (peano2, addeq)