Theorem slameqd ≪ | index | src | ≫

theorem slameqd (_G: wff) {x: nat} (_A1 _A2: set x):
  $ _G -> _A1 == _A2 $ >
  $ _G -> (\\ x, _A1) == (\\ x, _A2) $;

Step	Hyp	Ref	Expression
1		eqidd	_G -> snd p, y = snd p, y
2		eqidd	_G -> fst p = fst p
3		hyp _Ah	_G -> _A1 == _A2
4	2, 3	sbseqd	_G -> S[fst p / x] _A1 == S[fst p / x] _A2
5	1, 4	eleqd	_G -> (snd p, y e. S[fst p / x] _A1 <-> snd p, y e. S[fst p / x] _A2)
6	5	abeqd	_G -> {y \| snd p, y e. S[fst p / x] _A1} == {y \| snd p, y e. S[fst p / x] _A2}
7	6	sabeqd	_G -> S\ p, {y \| snd p, y e. S[fst p / x] _A1} == S\ p, {y \| snd p, y e. S[fst p / x] _A2}
8	7	conv slam	_G -> (\\ x, _A1) == (\\ x, _A2)

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)