Theorem writeeqd ≪ | index | src | ≫

theorem writeeqd (_G: wff) (_F1 _F2: set) (_a1 _a2 _b1 _b2: nat):
  $ _G -> _F1 == _F2 $ >
  $ _G -> _a1 = _a2 $ >
  $ _G -> _b1 = _b2 $ >
  $ _G -> write _F1 _a1 _b1 == write _F2 _a2 _b2 $;

Step	Hyp	Ref	Expression
1		hyp _ah	_G -> _a1 = _a2
2	1	eqeq2d	_G -> (x = _a1 <-> x = _a2)
3		hyp _bh	_G -> _b1 = _b2
4	3	eqeq2d	_G -> (y = _b1 <-> y = _b2)
5		hyp _Fh	_G -> _F1 == _F2
6	5	eleq2d	_G -> (x, y e. _F1 <-> x, y e. _F2)
7	2, 4, 6	ifpeqd	_G -> (ifp (x = _a1) (y = _b1) (x, y e. _F1) <-> ifp (x = _a2) (y = _b2) (x, y e. _F2))
8	7	abeqd	_G -> {y \| ifp (x = _a1) (y = _b1) (x, y e. _F1)} == {y \| ifp (x = _a2) (y = _b2) (x, y e. _F2)}
9	8	sabeqd	_G -> S\ x, {y \| ifp (x = _a1) (y = _b1) (x, y e. _F1)} == S\ x, {y \| ifp (x = _a2) (y = _b2) (x, y e. _F2)}
10	9	conv write	_G -> write _F1 _a1 _b1 == write _F2 _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)