Theorem boolext ≪ | index | src | ≫ 

theorem boolext (a b: nat):
  $ bool a -> bool b -> (true a <-> true b <-> a = b) $;
StepHypRefExpression
1 nattrue 
bool a -> nat (true a) = a
2 anll 
bool a /\ bool b /\ (true a <-> true b) -> bool a
3 1, 2 syl 
bool a /\ bool b /\ (true a <-> true b) -> nat (true a) = a
4 nattrue 
bool b -> nat (true b) = b
5 anlr 
bool a /\ bool b /\ (true a <-> true b) -> bool b
6 4, 5 syl 
bool a /\ bool b /\ (true a <-> true b) -> nat (true b) = b
7 3, 6 eqeqd 
bool a /\ bool b /\ (true a <-> true b) -> (nat (true a) = nat (true b) <-> a = b)
8 nateq 
(true a <-> true b) -> nat (true a) = nat (true b)
9 8 anwr 
bool a /\ bool b /\ (true a <-> true b) -> nat (true a) = nat (true b)
10 7, 9 mpbid 
bool a /\ bool b /\ (true a <-> true b) -> a = b
11 10 exp 
bool a /\ bool b -> (true a <-> true b) -> a = b
12 trueeq 
a = b -> (true a <-> true b)
13 12 a1i 
bool a /\ bool b -> a = b -> (true a <-> true b)
14 11, 13 ibid 
bool a /\ bool b -> (true a <-> true b <-> a = b)
15 14 exp 
bool a -> bool b -> (true a <-> true b <-> a = b)

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 (peano1, peano2, peano5, addeq, add0, addS)