Theorem imdistan 432

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		pm5.42 313	. 2 |- ( ( ph -> ( ps -> ch ) ) <-> ( ph -> ( ps -> ( ph /\ ch ) ) ) )
2		impexp 259	. 2 |- ( ( ( ph /\ ps ) -> ( ph /\ ch ) ) <-> ( ph -> ( ps -> ( ph /\ ch ) ) ) )
3	1, 2	bitr4i 185	1 |- ( ( ph -> ( ps -> ch ) ) <-> ( ( ph /\ ps ) -> ( ph /\ ch ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  imdistand  435  pm5.3  458  rmoim  2791  ss2rab  3070  bezoutlembi  10394