Theorem bi3and 636

Description: Deduction joining 3 equivalences to form equivalence of conjunctions.

Hypotheses

Ref	Expression
bi3d.1	|- (ph -> (ps <-> ch))
bi3d.2	|- (ph -> (th <-> ta ))
bi3d.3	|- (ph -> (et <-> ze))

Assertion

Ref	Expression
bi3and	|- (ph -> ((ps /\ th /\ et) <-> (ch /\ ta /\ ze)))

Proof of Theorem bi3and

Step	Hyp	Ref	Expression
1		bi3d.1	. . . 4 |- (ph -> (ps <-> ch))
2		bi3d.2	. . . 4 |- (ph -> (th <-> ta ))
3	1, 2	anbi12d 476	. . 3 |- (ph -> ((ps /\ th) <-> (ch /\ ta )))
4		bi3d.3	. . 3 |- (ph -> (et <-> ze))
5	3, 4	anbi12d 476	. 2 |- (ph -> (((ps /\ th) /\ et) <-> ((ch /\ ta ) /\ ze)))
6		df-3an 583	. 2 |- ((ps /\ th /\ et) <-> ((ps /\ th) /\ et))
7		df-3an 583	. 2 |- ((ch /\ ta /\ ze) <-> ((ch /\ ta ) /\ ze))
8	5, 6, 7	3bitr4g 428	1 |- (ph -> ((ps /\ th /\ et) <-> (ch /\ ta /\ ze)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   /\ w3a 581

This theorem is referenced by:  so 2152  limeq 2211  tz9.1 3490  mulcant2 4209  sqrlem20 4750

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  df-3an 583

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