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Theorem 3orim123d 1443
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 966 . . 3 (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))
4 3anim123d.3 . . 3 (𝜑 → (𝜂𝜁))
53, 4orim12d 966 . 2 (𝜑 → (((𝜓𝜃) ∨ 𝜂) → ((𝜒𝜏) ∨ 𝜁)))
6 df-3or 1087 . 2 ((𝜓𝜃𝜂) ↔ ((𝜓𝜃) ∨ 𝜂))
7 df-3or 1087 . 2 ((𝜒𝜏𝜁) ↔ ((𝜒𝜏) ∨ 𝜁))
85, 6, 73imtr4g 296 1 (𝜑 → ((𝜓𝜃𝜂) → (𝜒𝜏𝜁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 847  w3o 1085
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 207  df-an 396  df-or 848  df-3or 1087
This theorem is referenced by:  fr3nr  7790  soxp  8152  poxp3  8173  zorn2lem6  10538  fpwwe2lem11  10678  fpwwe2lem12  10679  sltres  27721  colinearalglem4  28938  chnso  32987  constrconj  33749  colinearxfr  36056  weiunso  36448  fin2so  37593  frege133d  43754  el1fzopredsuc  47274  fmtno4prmfac  47496
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