Metamath Proof Explorer

Theorem ifpbi13

Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020)

Ref Expression
Assertion ifpbi13 
|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( if- ( ph , ta , ch ) <-> if- ( ps , ta , th ) ) )

Proof

Step Hyp Ref Expression
1 simpl 
 |-  ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ph <-> ps ) )
2 1 imbi1d 
 |-  ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( ph -> ta ) <-> ( ps -> ta ) ) )
3 notbi 
 |-  ( ( ph <-> ps ) <-> ( -. ph <-> -. ps ) )
4 imbi12 
 |-  ( ( -. ph <-> -. ps ) -> ( ( ch <-> th ) -> ( ( -. ph -> ch ) <-> ( -. ps -> th ) ) ) )
5 3 4 sylbi 
 |-  ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( -. ph -> ch ) <-> ( -. ps -> th ) ) ) )
6 5 imp 
 |-  ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( -. ph -> ch ) <-> ( -. ps -> th ) ) )
7 2 6 anbi12d 
 |-  ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( ( ph -> ta ) /\ ( -. ph -> ch ) ) <-> ( ( ps -> ta ) /\ ( -. ps -> th ) ) ) )
8 dfifp2 
 |-  ( if- ( ph , ta , ch ) <-> ( ( ph -> ta ) /\ ( -. ph -> ch ) ) )
9 dfifp2 
 |-  ( if- ( ps , ta , th ) <-> ( ( ps -> ta ) /\ ( -. ps -> th ) ) )
10 7 8 9 3bitr4g 
 |-  ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( if- ( ph , ta , ch ) <-> if- ( ps , ta , th ) ) )