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Theorem 3orbi123d 1289
Description: Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.)
Hypotheses
Ref Expression
bi3d.1 (𝜑 → (𝜓𝜒))
bi3d.2 (𝜑 → (𝜃𝜏))
bi3d.3 (𝜑 → (𝜂𝜁))
Assertion
Ref Expression
3orbi123d (𝜑 → ((𝜓𝜃𝜂) ↔ (𝜒𝜏𝜁)))

Proof of Theorem 3orbi123d
StepHypRef Expression
1 bi3d.1 . . . 4 (𝜑 → (𝜓𝜒))
2 bi3d.2 . . . 4 (𝜑 → (𝜃𝜏))
31, 2orbi12d 782 . . 3 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
4 bi3d.3 . . 3 (𝜑 → (𝜂𝜁))
53, 4orbi12d 782 . 2 (𝜑 → (((𝜓𝜃) ∨ 𝜂) ↔ ((𝜒𝜏) ∨ 𝜁)))
6 df-3or 963 . 2 ((𝜓𝜃𝜂) ↔ ((𝜓𝜃) ∨ 𝜂))
7 df-3or 963 . 2 ((𝜒𝜏𝜁) ↔ ((𝜒𝜏) ∨ 𝜁))
85, 6, 73bitr4g 222 1 (𝜑 → ((𝜓𝜃𝜂) ↔ (𝜒𝜏𝜁)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 697  w3o 961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698
This theorem depends on definitions:  df-bi 116  df-3or 963
This theorem is referenced by:  ordtriexmid  4437  wetriext  4491  nntri3or  6389  tridc  6793  ltsopi  7135  pitri3or  7137  nqtri3or  7211  elz  9063  ztri3or  9104  qtri3or  10027  trilpo  13266
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