Theorem casesifp 1026

Description: Version of cases 992 expressed using if-. Case disjunction according to the value of ph. One can see this as a proof that the two hypotheses characterize the conditional operator for propositions. For the converses, see ifptru 1023 and ifpfal 1024. (Contributed by BJ, 20-Sep-2019.)

Hypotheses

Ref	Expression
casesifp.1	|- ( ph -> ( ps <-> ch ) )
casesifp.2	|- ( -. ph -> ( ps <-> th ) )

Assertion

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

Proof of Theorem casesifp

Step	Hyp	Ref	Expression
1		casesifp.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2		casesifp.2	. . 3 |- ( -. ph -> ( ps <-> th ) )
3	1, 2	cases 992	. 2 |- ( ps <-> ( ( ph /\ ch ) \/ ( -. ph /\ th ) ) )
4		df-ifp 1013	. 2 |- (if- ( ph , ch , th ) <-> ( ( ph /\ ch ) \/ ( -. ph /\ th ) ) )
5	3, 4	bitr4i 267	1 |- ( ps <-> if- ( ph , ch , th ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ 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:  hadifp  1544  cadifp  1557