Theorem imdistan 442

Description: Distribution of implication with conjunction.

Assertion

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

Proof of Theorem imdistan

Step	Hyp	Ref	Expression
1		anc2l 300	. . 3 |- ((ph -> (ps -> ch)) -> (ph -> (ps -> (ph /\ ch))))
2	1	imp3a 361	. 2 |- ((ph -> (ps -> ch)) -> ((ph /\ ps) -> (ph /\ ch)))
3		pm3.27 323	. . . 4 |- ((ph /\ ch) -> ch)
4	3	imim2i 17	. . 3 |- (((ph /\ ps) -> (ph /\ ch)) -> ((ph /\ ps) -> ch))
5	4	exp3a 375	. 2 |- (((ph /\ ps) -> (ph /\ ch)) -> (ph -> (ps -> ch)))
6	2, 5	impbi 157	1 |- ((ph -> (ps -> ch)) <-> ((ph /\ ps) -> (ph /\ ch)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223

This theorem is referenced by:  imdistand 445  r19.22 1731  ss2rab 2123

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225

Copyright terms: Public domain