Theorem ifpneg | index | src |

pub theorem ifpneg (p a b: wff): $ ~p -> (ifp p a b <-> b) $;
StepHypRefExpression
1 bior1
~(p /\ a) -> (p /\ a \/ ~p /\ b <-> ~p /\ b)
2 1 conv ifp
~(p /\ a) -> (ifp p a b <-> ~p /\ b)
3 con3
(p /\ a -> p) -> ~p -> ~(p /\ a)
4 anl
p /\ a -> p
5 3, 4 ax_mp
~p -> ~(p /\ a)
6 2, 5 syl
~p -> (ifp p a b <-> ~p /\ b)
7 bian1
~p -> (~p /\ b <-> b)
8 6, 7 bitrd
~p -> (ifp p a b <-> b)

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp)