Theorem ifpdfan 37810

Description: Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.)

Assertion

Ref	Expression
ifpdfan	|- ( ( ph /\ ps ) <-> if- ( ph , ps , F. ) )

Proof of Theorem ifpdfan

Step	Hyp	Ref	Expression
1		fal 1490	. . . 4 |- -. F.
2	1	intnan 960	. . 3 |- -. ( -. ph /\ F. )
3	2	biorfi 422	. 2 |- ( ( ph /\ ps ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ F. ) ) )
4		df-ifp 1013	. 2 |- (if- ( ph , ps , F. ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ F. ) ) )
5	3, 4	bitr4i 267	1 |- ( ( ph /\ ps ) <-> if- ( ph , ps , F. ) )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384  if-wif 1012   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:  ifpdfnan  37831  ifpdfxor  37832