Theorem ifbothd ≪ | index | src | ≫ 

theorem ifbothd (G: wff) (a b: nat) (p q qa qb: wff):
  $ if p a b = a -> (q <-> qa) $ >
  $ if p a b = b -> (q <-> qb) $ >
  $ G /\ p -> qa $ >
  $ G /\ ~p -> qb $ >
  $ G -> q $;
StepHypRefExpression
1 hyp ea 
if p a b = a -> (q <-> qa)
2 ifpos 
p -> if p a b = a
3 2 anwr 
G /\ p -> if p a b = a
4 1, 3 syl 
G /\ p -> (q <-> qa)
5 hyp h1 
G /\ p -> qa
6 4, 5 mpbird 
G /\ p -> q
7 hyp eb 
if p a b = b -> (q <-> qb)
8 ifneg 
~p -> if p a b = b
9 8 anwr 
G /\ ~p -> if p a b = b
10 7, 9 syl 
G /\ ~p -> (q <-> qb)
11 hyp h2 
G /\ ~p -> qb
12 10, 11 mpbird 
G /\ ~p -> q
13 6, 12 casesda 
G -> q

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)