Metamath Proof Explorer


Theorem ifpfal

Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse . This is essentially dedlemb . (Contributed by BJ, 20-Sep-2019) (Proof shortened by Wolf Lammen, 25-Jun-2020)

Ref Expression
Assertion ifpfal
|- ( -. ph -> ( if- ( ph , ps , ch ) <-> ch ) )

Proof

Step Hyp Ref Expression
1 ifpn
 |-  ( if- ( ph , ps , ch ) <-> if- ( -. ph , ch , ps ) )
2 ifptru
 |-  ( -. ph -> ( if- ( -. ph , ch , ps ) <-> ch ) )
3 1 2 syl5bb
 |-  ( -. ph -> ( if- ( ph , ps , ch ) <-> ch ) )