Theorem imdistan 448

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		pm5.42 320	. 2 |- ( ( ph -> ( ps -> ch ) ) <-> ( ph -> ( ps -> ( ph /\ ch ) ) ) )
2		impexp 263	. 2 |- ( ( ( ph /\ ps ) -> ( ph /\ ch ) ) <-> ( ph -> ( ps -> ( ph /\ ch ) ) ) )
3	1, 2	bitr4i 187	1 |- ( ( ph -> ( ps -> ch ) ) <-> ( ( ph /\ ps ) -> ( ph /\ ch ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  imdistand  451  pm5.3  479  rmoim  3027  ss2rab  3324  bezoutlembi  12782