Theorem neeqd ≪ | index | src | ≫

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


_G -> _a1 = _a2
_G -> _b1 = _b2
_G -> (_a1 = _b1 <-> _a2 = _b2)
_G -> (~_a1 = _b1 <-> ~_a2 = _b2)
_G -> (_a1 != _b1 <-> _a2 != _b2)

Step	Hyp	Ref	Expression
1		hyp _ah	_G -> _a1 = _a2
2		hyp _bh	_G -> _b1 = _b2
3	1, 2	eqeqd	_G -> (_a1 = _b1 <-> _a2 = _b2)
4	3	noteqd	_G -> (~_a1 = _b1 <-> ~_a2 = _b2)
5	4	conv ne	_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)