Theorem imdistan 725

Description: Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.)

Assertion

Ref	Expression
imdistan	|- ( ( ph -> ( ps -> ch ) ) <-> ( ( ph /\ ps ) -> ( ph /\ ch ) ) )

Proof of Theorem imdistan

Step	Hyp	Ref	Expression
1		pm5.42 571	. 2 |- ( ( ph -> ( ps -> ch ) ) <-> ( ph -> ( ps -> ( ph /\ ch ) ) ) )
2		impexp 462	. 2 |- ( ( ( ph /\ ps ) -> ( ph /\ ch ) ) <-> ( ph -> ( ps -> ( ph /\ ch ) ) ) )
3	1, 2	bitr4i 267	1 |- ( ( ph -> ( ps -> ch ) ) <-> ( ( ph /\ ps ) -> ( ph /\ ch ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384

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-an 386

This theorem is referenced by:  imdistand  728  pm5.3  748  rmoim  3407  ss2rab  3678  marypha2lem3  8343  inxpss3  34084