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Theorem 3orbi123i 1133
Description: Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
Hypotheses
Ref Expression
bi3.1 (𝜑𝜓)
bi3.2 (𝜒𝜃)
bi3.3 (𝜏𝜂)
Assertion
Ref Expression
3orbi123i ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))

Proof of Theorem 3orbi123i
StepHypRef Expression
1 bi3.1 . . . 4 (𝜑𝜓)
2 bi3.2 . . . 4 (𝜒𝜃)
31, 2orbi12i 716 . . 3 ((𝜑𝜒) ↔ (𝜓𝜃))
4 bi3.3 . . 3 (𝜏𝜂)
53, 4orbi12i 716 . 2 (((𝜑𝜒) ∨ 𝜏) ↔ ((𝜓𝜃) ∨ 𝜂))
6 df-3or 925 . 2 ((𝜑𝜒𝜏) ↔ ((𝜑𝜒) ∨ 𝜏))
7 df-3or 925 . 2 ((𝜓𝜃𝜂) ↔ ((𝜓𝜃) ∨ 𝜂))
85, 6, 73bitr4i 210 1 ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
Colors of variables: wff set class
Syntax hints:  wb 103  wo 664  w3o 923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665
This theorem depends on definitions:  df-bi 115  df-3or 925
This theorem is referenced by:  nnwetri  6606
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