Theorem subsneqd ≪ | index | src | ≫ 

theorem subsneqd (_G: wff) (_A1 _A2: set):
  $ _G -> _A1 == _A2 $ >
  $ _G -> (subsn _A1 <-> subsn _A2) $;
StepHypRefExpression
1 eqidd 
_G -> x = x
2 hyp _Ah 
_G -> _A1 == _A2
3 1, 2 eleqd 
_G -> (x e. _A1 <-> x e. _A2)
4 eqidd 
_G -> y = y
5 4, 2 eleqd 
_G -> (y e. _A1 <-> y e. _A2)
6 biidd 
_G -> (x = y <-> x = y)
7 5, 6 imeqd 
_G -> (y e. _A1 -> x = y <-> y e. _A2 -> x = y)
8 3, 7 imeqd 
_G -> (x e. _A1 -> y e. _A1 -> x = y <-> x e. _A2 -> y e. _A2 -> x = y)
9 8 aleqd 
_G -> (A. y (x e. _A1 -> y e. _A1 -> x = y) <-> A. y (x e. _A2 -> y e. _A2 -> x = y))
10 9 aleqd 
_G -> (A. x A. y (x e. _A1 -> y e. _A1 -> x = y) <-> A. x A. y (x e. _A2 -> y e. _A2 -> x = y))
11 10 conv subsn 
_G -> (subsn _A1 <-> subsn _A2)

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)