Theorem ifpimpda 1028

Description: Separation of the values of the conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 27-Feb-2021.)

Hypotheses

Ref	Expression
ifpimpda.1	|- ( ( ph /\ ps ) -> ch )
ifpimpda.2	|- ( ( ph /\ -. ps ) -> th )

Assertion

Ref	Expression
ifpimpda	|- ( ph -> if- ( ps , ch , th ) )

Proof of Theorem ifpimpda

Step	Hyp	Ref	Expression
1		ifpimpda.1	. . 3 |- ( ( ph /\ ps ) -> ch )
2	1	ex 450	. 2 |- ( ph -> ( ps -> ch ) )
3		ifpimpda.2	. . 3 |- ( ( ph /\ -. ps ) -> th )
4	3	ex 450	. 2 |- ( ph -> ( -. ps -> th ) )
5		dfifp2 1014	. 2 |- (if- ( ps , ch , th ) <-> ( ( ps -> ch ) /\ ( -. ps -> th ) ) )
6	2, 4, 5	sylanbrc 698	1 |- ( ph -> if- ( ps , ch , th ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384  if-wif 1012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013

This theorem is referenced by:  ifpprsnss  4299  wlkp1lem8  26577  1wlkdlem4  27000