Theorem isfeqd ≪ | index | src | ≫ 

theorem isfeqd (_G: wff) (_A1 _A2: set):
  $ _G -> _A1 == _A2 $ >
  $ _G -> (isfun _A1 <-> isfun _A2) $;
StepHypRefExpression
1 hyp _Ah 
_G -> _A1 == _A2
2 1 eleq2d 
_G -> (a, b e. _A1 <-> a, b e. _A2)
3 1 eleq2d 
_G -> (a, b2 e. _A1 <-> a, b2 e. _A2)
4 3 imeq1d 
_G -> (a, b2 e. _A1 -> b = b2 <-> a, b2 e. _A2 -> b = b2)
5 2, 4 imeqd 
_G -> (a, b e. _A1 -> a, b2 e. _A1 -> b = b2 <-> a, b e. _A2 -> a, b2 e. _A2 -> b = b2)
6 5 aleqd 
_G -> (A. b2 (a, b e. _A1 -> a, b2 e. _A1 -> b = b2) <-> A. b2 (a, b e. _A2 -> a, b2 e. _A2 -> b = b2))
7 6 aleqd 
_G -> (A. b A. b2 (a, b e. _A1 -> a, b2 e. _A1 -> b = b2) <-> A. b A. b2 (a, b e. _A2 -> a, b2 e. _A2 -> b = b2))
8 7 aleqd 
_G -> (A. a A. b A. b2 (a, b e. _A1 -> a, b2 e. _A1 -> b = b2) <-> A. a A. b A. b2 (a, b e. _A2 -> a, b2 e. _A2 -> b = b2))
9 8 conv isfun 
_G -> (isfun _A1 <-> isfun _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)