Theorem bi3ord 635

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

Hypotheses

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

Assertion

Ref	Expression
bi3ord	|- (ph -> ((ps \/ th \/ et) <-> (ch \/ ta \/ ze)))

Proof of Theorem bi3ord

Step	Hyp	Ref	Expression
1		bi3d.1	. . . 4 |- (ph -> (ps <-> ch))
2		bi3d.2	. . . 4 |- (ph -> (th <-> ta ))
3	1, 2	orbi12d 475	. . 3 |- (ph -> ((ps \/ th) <-> (ch \/ ta )))
4		bi3d.3	. . 3 |- (ph -> (et <-> ze))
5	3, 4	orbi12d 475	. 2 |- (ph -> (((ps \/ th) \/ et) <-> ((ch \/ ta ) \/ ze)))
6		df-3or 582	. 2 |- ((ps \/ th \/ et) <-> ((ps \/ th) \/ et))
7		df-3or 582	. 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   \/ wo 195   \/ w3o 580

This theorem is referenced by:  moeq3 1432  soeq1 2141  solin 2145  dfwe2 2187  weinxp 2467  isowe 2941  f1oweOLD 2944  rdgeq1 2972  rdgeq2 2973  rdglem2 2976  ltsopr 3930  elz 4565

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-or 197  df-an 198  df-3or 582

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