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Theorem 3anim123d 1439
Description: Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.)
Hypotheses
Ref Expression
3anim123d.1 (𝜑 → (𝜓𝜒))
3anim123d.2 (𝜑 → (𝜃𝜏))
3anim123d.3 (𝜑 → (𝜂𝜁))
Assertion
Ref Expression
3anim123d (𝜑 → ((𝜓𝜃𝜂) → (𝜒𝜏𝜁)))

Proof of Theorem 3anim123d
StepHypRef Expression
1 3anim123d.1 . . . 4 (𝜑 → (𝜓𝜒))
2 3anim123d.2 . . . 4 (𝜑 → (𝜃𝜏))
31, 2anim12d 607 . . 3 (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))
4 3anim123d.3 . . 3 (𝜑 → (𝜂𝜁))
53, 4anim12d 607 . 2 (𝜑 → (((𝜓𝜃) ∧ 𝜂) → ((𝜒𝜏) ∧ 𝜁)))
6 df-3an 1086 . 2 ((𝜓𝜃𝜂) ↔ ((𝜓𝜃) ∧ 𝜂))
7 df-3an 1086 . 2 ((𝜒𝜏𝜁) ↔ ((𝜒𝜏) ∧ 𝜁))
85, 6, 73imtr4g 295 1 (𝜑 → ((𝜓𝜃𝜂) → (𝜒𝜏𝜁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084
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 206  df-an 395  df-3an 1086
This theorem is referenced by:  pofun  5608  isopolem  7352  issmo2  8370  smores  8373  inawina  10715  gchina  10724  repswcshw  14798  coprmprod  16635  issubmnd  18724  issubg2  19104  issubrng2  20507  issubrg2  20543  rnglidlmsgrp  21153  rnglidlrng  21154  ocv2ss  21622  issubassa3  21816  sslm  23247  cmetcaulem  25260  axcontlem4  28850  axcontlem8  28854  redwlk  29558  clwwlknwwlksn  29920  numclwwlk1lem2foa  30236  dipsubdir  30730  subgrpth  34875  cgr3tr4  35779  idinside  35811  ftc1anclem7  37303  fzmul  37345  fdc1  37350  rngosubdi  37549  rngosubdir  37550  cdlemg33a  40309  upwlkwlk  47387
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