Theorem lmemeqd | index | src |

theorem lmemeqd (_G: wff) (_a1 _a2 _l1 _l2: nat):
  $ _G -> _a1 = _a2 $ >
  $ _G -> _l1 = _l2 $ >
  $ _G -> (_a1 IN _l1 <-> _a2 IN _l2) $;
StepHypRefExpression
1 hyp _ah
_G -> _a1 = _a2
2 hyp _lh
_G -> _l1 = _l2
3 2 lmemseqd
_G -> lmems _l1 = lmems _l2
4 3 nseqd
_G -> lmems _l1 == lmems _l2
5 1, 4 eleqd
_G -> (_a1 e. lmems _l1 <-> _a2 e. lmems _l2)
6 5 conv lmem
_G -> (_a1 IN _l1 <-> _a2 IN _l2)

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), axs_peano (peano2, addeq, muleq)