Theorem 3anim123d 898

Description: Deduction joining 3 implications to form implication of conjunctions.

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 557	. . 3 |- (ph -> ((ps /\ th) -> (ch /\ ta)))
4		3anim123d.3	. . 3 |- (ph -> (et -> ze))
5	3, 4	anim12d 557	. 2 |- (ph -> (((ps /\ th) /\ et) -> ((ch /\ ta) /\ ze)))
6		df-3an 776	. 2 |- ((ps /\ th /\ et) <-> ((ps /\ th) /\ et))
7		df-3an 776	. 2 |- ((ch /\ ta /\ ze) <-> ((ch /\ ta) /\ ze))
8	5, 6, 7	3imtr4g 552	1 |- (ph -> ((ps /\ th /\ et) -> (ch /\ ta /\ ze)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774

This theorem is referenced by:  poss 2836  abfii4 4544  expord2t 6543  opni3 7818  grpnnncan2 8043  ipsubdir 8452  hmphsyma 10451  cmpassoh 10609

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 776

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