Theorem sseqd ≪ | index | src | ≫ 

theorem sseqd (_G: wff) (_A1 _A2 _B1 _B2: set):
  $ _G -> _A1 == _A2 $ >
  $ _G -> _B1 == _B2 $ >
  $ _G -> (_A1 C_ _B1 <-> _A2 C_ _B2) $;
StepHypRefExpression
1 eqidd 
_G -> x = x
2 hyp _Ah 
_G -> _A1 == _A2
3 1, 2 eleqd 
_G -> (x e. _A1 <-> x e. _A2)
4 hyp _Bh 
_G -> _B1 == _B2
5 1, 4 eleqd 
_G -> (x e. _B1 <-> x e. _B2)
6 3, 5 imeqd 
_G -> (x e. _A1 -> x e. _B1 <-> x e. _A2 -> x e. _B2)
7 6 aleqd 
_G -> (A. x (x e. _A1 -> x e. _B1) <-> A. x (x e. _A2 -> x e. _B2))
8 7 conv subset 
_G -> (_A1 C_ _B1 <-> _A2 C_ _B2)

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_12), axs_set (ax_8)