Theorem 3anandis 920

Description: Inference that undistributes a triple conjunction in the antecedent.

Hypothesis

Ref	Expression
3anandis.1	|- (((ph /\ ps) /\ (ph /\ ch) /\ (ph /\ th)) -> ta)

Assertion

Ref	Expression
3anandis	|- ((ph /\ (ps /\ ch /\ th)) -> ta)

Proof of Theorem 3anandis

Step	Hyp	Ref	Expression
1		3anandis.1	. 2 |- (((ph /\ ps) /\ (ph /\ ch) /\ (ph /\ th)) -> ta)
2		3simp1 788	. . 3 |- ((ps /\ ch /\ th) -> ps)
3	2	anim2i 335	. 2 |- ((ph /\ (ps /\ ch /\ th)) -> (ph /\ ps))
4		3simp2 789	. . 3 |- ((ps /\ ch /\ th) -> ch)
5	4	anim2i 335	. 2 |- ((ph /\ (ps /\ ch /\ th)) -> (ph /\ ch))
6		3simp3 790	. . 3 |- ((ps /\ ch /\ th) -> th)
7	6	anim2i 335	. 2 |- ((ph /\ (ps /\ ch /\ th)) -> (ph /\ th))
8	1, 3, 5, 7	syl3anc 858	1 |- ((ph /\ (ps /\ ch /\ th)) -> ta)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775

This theorem is referenced by:  hcau2 9055

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  df-3an 777

Copyright terms: Public domain