Theorem imdistan 339

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 248	. . 3 |- ((ph -> (ps -> ch)) -> (ph -> (ps -> (ph /\ ch))))
2	1	imp3a 279	. 2 |- ((ph -> (ps -> ch)) -> ((ph /\ ps) -> (ph /\ ch)))
3		pm3.27 260	. . . 4 |- ((ph /\ ch) -> ch)
4	3	syl3 18	. . 3 |- (((ph /\ ps) -> (ph /\ ch)) -> ((ph /\ ps) -> ch))
5	4	exp3a 292	. 2 |- (((ph /\ ps) -> (ph /\ ch)) -> (ph -> (ps -> ch)))
6	2, 5	impbi 139	1 |- ((ph -> (ps -> ch)) <-> ((ph /\ ps) -> (ph /\ ch)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196

This theorem is referenced by:  imdistand 342  r19.22 1272  ss2rab 1553

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

This theorem depends on definitions:  df-bi 128  df-an 198

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