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Theorem 3anim123d 1555
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 587 . . 3 (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))
4 3anim123d.3 . . 3 (𝜑 → (𝜂𝜁))
53, 4anim12d 587 . 2 (𝜑 → (((𝜓𝜃) ∧ 𝜂) → ((𝜒𝜏) ∧ 𝜁)))
6 df-3an 1074 . 2 ((𝜓𝜃𝜂) ↔ ((𝜓𝜃) ∧ 𝜂))
7 df-3an 1074 . 2 ((𝜒𝜏𝜁) ↔ ((𝜒𝜏) ∧ 𝜁))
85, 6, 73imtr4g 285 1 (𝜑 → ((𝜓𝜃𝜂) → (𝜒𝜏𝜁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072
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-an 385  df-3an 1074
This theorem is referenced by:  pofun  5204  isopolem  6760  issmo2  7617  smores  7620  inawina  9725  gchina  9734  repswcshw  13779  coprmprod  15598  issubmnd  17540  issubg2  17831  issubrg2  19023  ocv2ss  20240  sslm  21326  cmetcaulem  23307  axcontlem4  26068  axcontlem8  26072  redwlk  26801  clwwlknwwlksn  27188  clwwlknwwlksnOLD  27189  clwwlknonwwlknonb  27276  numclwlk1lem2foa  27535  dipsubdir  28034  cgr3tr4  32487  idinside  32519  ftc1anclem7  33823  fzmul  33869  fdc1  33874  rngosubdi  34076  rngosubdir  34077  cdlemg33a  36515  upwlkwlk  42249  lidlmsgrp  42455  lidlrng  42456
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