Theorem ifeqd ≪ | index | src | ≫

theorem ifeqd (_G _p1 _p2: wff) (_a1 _a2 _b1 _b2: nat):
  $ _G -> (_p1 <-> _p2) $ >
  $ _G -> _a1 = _a2 $ >
  $ _G -> _b1 = _b2 $ >
  $ _G -> if _p1 _a1 _b1 = if _p2 _a2 _b2 $;

Step	Hyp	Ref	Expression
1		hyp _ph	_G -> (_p1 <-> _p2)
2		eqidd	_G -> n = n
3		hyp _ah	_G -> _a1 = _a2
4	2, 3	eqeqd	_G -> (n = _a1 <-> n = _a2)
5		hyp _bh	_G -> _b1 = _b2
6	2, 5	eqeqd	_G -> (n = _b1 <-> n = _b2)
7	1, 4, 6	ifpeqd	_G -> (ifp _p1 (n = _a1) (n = _b1) <-> ifp _p2 (n = _a2) (n = _b2))
8	7	abeqd	_G -> {n \| ifp _p1 (n = _a1) (n = _b1)} == {n \| ifp _p2 (n = _a2) (n = _b2)}
9	8	theeqd	_G -> the {n \| ifp _p1 (n = _a1) (n = _b1)} = the {n \| ifp _p2 (n = _a2) (n = _b2)}
10	9	conv if	_G -> if _p1 _a1 _b1 = if _p2 _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)