Theorem jcab 858

Description: Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 27-Nov-2013.)

Assertion

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

Proof of Theorem jcab

Step	Hyp	Ref	Expression
1		simpl 457	. . . 4 |- ( ( ps /\ ch ) -> ps )
2	1	imim2i 14	. . 3 |- ( ( ph -> ( ps /\ ch ) ) -> ( ph -> ps ) )
3		simpr 461	. . . 4 |- ( ( ps /\ ch ) -> ch )
4	3	imim2i 14	. . 3 |- ( ( ph -> ( ps /\ ch ) ) -> ( ph -> ch ) )
5	2, 4	jca 532	. 2 |- ( ( ph -> ( ps /\ ch ) ) -> ( ( ph -> ps ) /\ ( ph -> ch ) ) )
6		pm3.43 857	. 2 |- ( ( ( ph -> ps ) /\ ( ph -> ch ) ) -> ( ph -> ( ps /\ ch ) ) )
7	5, 6	impbii 188	1 |- ( ( ph -> ( ps /\ ch ) ) <-> ( ( ph -> ps ) /\ ( ph -> ch ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369

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 185  df-an 371

This theorem is referenced by:  ordi  859  pm4.76  861  pm5.44  902  2mo2  2367  2eu4OLD  2377  ssconb  3592  ssin  3675  tfr3  6963  isprm2  13884  lgsquad2lem2  22826  ostthlem2  23005  2reu4a  30156  pclclN  33854