Theorem Ifeq ≪ | index | src | ≫ 

theorem Ifeq (_p1 _p2: wff) (_A1 _A2 _B1 _B2: set):
  $ (_p1 <-> _p2) ->
    _A1 == _A2 ->
    _B1 == _B2 ->
    If _p1 _A1 _B1 == If _p2 _A2 _B2 $;
StepHypRefExpression
1 anl 
(_p1 <-> _p2) /\ _A1 == _A2 -> (_p1 <-> _p2)
2 1 anwl 
(_p1 <-> _p2) /\ _A1 == _A2 /\ _B1 == _B2 -> (_p1 <-> _p2)
3 anr 
(_p1 <-> _p2) /\ _A1 == _A2 -> _A1 == _A2
4 3 anwl 
(_p1 <-> _p2) /\ _A1 == _A2 /\ _B1 == _B2 -> _A1 == _A2
5 anr 
(_p1 <-> _p2) /\ _A1 == _A2 /\ _B1 == _B2 -> _B1 == _B2
6 2, 4, 5 Ifeqd 
(_p1 <-> _p2) /\ _A1 == _A2 /\ _B1 == _B2 -> If _p1 _A1 _B1 == If _p2 _A2 _B2
7 6 exp 
(_p1 <-> _p2) /\ _A1 == _A2 -> _B1 == _B2 -> If _p1 _A1 _B1 == If _p2 _A2 _B2
8 7 exp 
(_p1 <-> _p2) -> _A1 == _A2 -> _B1 == _B2 -> If _p1 _A1 _B1 == If _p2 _A2 _B2

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)