Metamath Proof Explorer

Theorem dfifp5

Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019)

Ref Expression
Assertion dfifp5 
|- ( if- ( ph , ps , ch ) <-> ( ( -. ph \/ ps ) /\ ( -. ph -> ch ) ) )

Proof

Step Hyp Ref Expression
1 dfifp2 
 |-  ( if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) )
2 imor 
 |-  ( ( ph -> ps ) <-> ( -. ph \/ ps ) )
3 2 anbi1i 
 |-  ( ( ( ph -> ps ) /\ ( -. ph -> ch ) ) <-> ( ( -. ph \/ ps ) /\ ( -. ph -> ch ) ) )
4 1 3 bitri 
 |-  ( if- ( ph , ps , ch ) <-> ( ( -. ph \/ ps ) /\ ( -. ph -> ch ) ) )