Theorem 3anim123d 1213

Description: Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.)

Hypotheses

Ref	Expression
3anim123d.1	|- (ph -> (ps -> ch))
3anim123d.2	|- (ph -> (th -> ta))
3anim123d.3	|- (ph -> (et -> ze))

Assertion

Ref	Expression
3anim123d	|- (ph -> ((ps /\ th /\ et) -> (ch /\ ta /\ ze)))

Proof of Theorem 3anim123d

Step	Hyp	Ref	Expression
1		3anim123d.1	. . . 4 |- (ph -> (ps -> ch))
2		3anim123d.2	. . . 4 |- (ph -> (th -> ta))
3	1, 2	anim12d 318	. . 3 |- (ph -> ((ps /\ th) -> (ch /\ ta)))
4		3anim123d.3	. . 3 |- (ph -> (et -> ze))
5	3, 4	anim12d 318	. 2 |- (ph -> (((ps /\ th) /\ et) -> ((ch /\ ta) /\ ze)))
6		df-3an 886	. 2 |- ((ps /\ th /\ et) <-> ((ps /\ th) /\ et))
7		df-3an 886	. 2 |- ((ch /\ ta /\ ze) <-> ((ch /\ ta) /\ ze))
8	5, 6, 7	3imtr4g 194	1 |- (ph -> ((ps /\ th /\ et) -> (ch /\ ta /\ ze)))

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 884

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-3an 886

This theorem is referenced by:  hb3and  1376  pofun  4040  soss  4042  isopolem  5404  isosolem  5406  issmo2  5845  smores  5848