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Theorem 3orim123d 1556
Description: Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.)
Hypotheses
Ref Expression
3anim123d.1 (𝜑 → (𝜓𝜒))
3anim123d.2 (𝜑 → (𝜃𝜏))
3anim123d.3 (𝜑 → (𝜂𝜁))
Assertion
Ref Expression
3orim123d (𝜑 → ((𝜓𝜃𝜂) → (𝜒𝜏𝜁)))

Proof of Theorem 3orim123d
StepHypRef Expression
1 3anim123d.1 . . . 4 (𝜑 → (𝜓𝜒))
2 3anim123d.2 . . . 4 (𝜑 → (𝜃𝜏))
31, 2orim12d 919 . . 3 (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))
4 3anim123d.3 . . 3 (𝜑 → (𝜂𝜁))
53, 4orim12d 919 . 2 (𝜑 → (((𝜓𝜃) ∨ 𝜂) → ((𝜒𝜏) ∨ 𝜁)))
6 df-3or 1073 . 2 ((𝜓𝜃𝜂) ↔ ((𝜓𝜃) ∨ 𝜂))
7 df-3or 1073 . 2 ((𝜒𝜏𝜁) ↔ ((𝜒𝜏) ∨ 𝜁))
85, 6, 73imtr4g 285 1 (𝜑 → ((𝜓𝜃𝜂) → (𝜒𝜏𝜁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  w3o 1071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073
This theorem is referenced by:  fr3nr  7144  soxp  7458  zorn2lem6  9515  fpwwe2lem12  9655  fpwwe2lem13  9656  colinearalglem4  25988  sltres  32121  colinearxfr  32488  fin2so  33709  frege133d  38559  el1fzopredsuc  41845  fmtno4prmfac  41994
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