Theorem jcab 567

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 107	. . . 4 |- ( ( ps /\ ch ) -> ps )
2	1	imim2i 12	. . 3 |- ( ( ph -> ( ps /\ ch ) ) -> ( ph -> ps ) )
3		simpr 108	. . . 4 |- ( ( ps /\ ch ) -> ch )
4	3	imim2i 12	. . 3 |- ( ( ph -> ( ps /\ ch ) ) -> ( ph -> ch ) )
5	2, 4	jca 300	. 2 |- ( ( ph -> ( ps /\ ch ) ) -> ( ( ph -> ps ) /\ ( ph -> ch ) ) )
6		pm3.43 566	. 2 |- ( ( ( ph -> ps ) /\ ( ph -> ch ) ) -> ( ph -> ( ps /\ ch ) ) )
7	5, 6	impbii 124	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:  pm4.76  568  pm5.44  867  2eu4  2034  ssconb  3105  ssin  3188  raaan  3347  tfri3  5976  isprm2  10499