Theorem ifpnot 37814

Description: Restate negated wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)

Assertion

Ref	Expression
ifpnot	|- ( -. ph <-> if- ( ph , F. , T. ) )

Proof of Theorem ifpnot

Step	Hyp	Ref	Expression
1		tru 1487	. . . 4 |- T.
2	1	olci 406	. . 3 |- ( ph \/ T. )
3	2	biantru 526	. 2 |- ( ( -. ph \/ F. ) <-> ( ( -. ph \/ F. ) /\ ( ph \/ T. ) ) )
4		fal 1490	. . 3 |- -. F.
5	4	biorfi 422	. 2 |- ( -. ph <-> ( -. ph \/ F. ) )
6		dfifp4 1016	. 2 |- (if- ( ph , F. , T. ) <-> ( ( -. ph \/ F. ) /\ ( ph \/ T. ) ) )
7	3, 5, 6	3bitr4i 292	1 |- ( -. ph <-> if- ( ph , F. , T. ) )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384  if-wif 1012   T. wtru 1484   F. wfal 1488

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  df-tru 1486  df-fal 1489

This theorem is referenced by: (None)