Theorem sbneqd | index | src |

theorem sbneqd (_G: wff) {x: nat} (_a1 _a2 _b1 _b2: nat x):
  $ _G -> _a1 = _a2 $ >
  $ _G -> _b1 = _b2 $ >
  $ _G -> N[_a1 / x] _b1 = N[_a2 / x] _b2 $;
StepHypRefExpression
1 hyp _ah
_G -> _a1 = _a2
2 eqidd
_G -> y = y
3 hyp _bh
_G -> _b1 = _b2
4 2, 3 eqeqd
_G -> (y = _b1 <-> y = _b2)
5 1, 4 sbeqd
_G -> ([_a1 / x] y = _b1 <-> [_a2 / x] y = _b2)
6 5 abeqd
_G -> {y | [_a1 / x] y = _b1} == {y | [_a2 / x] y = _b2}
7 6 theeqd
_G -> the {y | [_a1 / x] y = _b1} = the {y | [_a2 / x] y = _b2}
8 7 conv sbn
_G -> N[_a1 / x] _b1 = N[_a2 / x] _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)