Theorem Ifeq3a ≪ | index | src | ≫ 

theorem Ifeq3a (A B C: set) (p: wff):
  $ (~p -> B == C) -> If p A B == If p A C $;
StepHypRefExpression
1 Ifpos 
p -> If p A B == A
2 Ifpos 
p -> If p A C == A
3 1, 2 eqstr4d 
p -> If p A B == If p A C
4 3 a1i 
(~p -> B == C) -> p -> If p A B == If p A C
5 Ifeq3 
B == C -> If p A B == If p A C
6 5 imim2i 
(~p -> B == C) -> ~p -> If p A B == If p A C
7 4, 6 casesd 
(~p -> B == C) -> If p A B == If p A C

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)