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Theorem 3anim123d 1434
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 608 . . 3 (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))
4 3anim123d.3 . . 3 (𝜑 → (𝜂𝜁))
53, 4anim12d 608 . 2 (𝜑 → (((𝜓𝜃) ∧ 𝜂) → ((𝜒𝜏) ∧ 𝜁)))
6 df-3an 1081 . 2 ((𝜓𝜃𝜂) ↔ ((𝜓𝜃) ∧ 𝜂))
7 df-3an 1081 . 2 ((𝜒𝜏𝜁) ↔ ((𝜒𝜏) ∧ 𝜁))
85, 6, 73imtr4g 297 1 (𝜑 → ((𝜓𝜃𝜂) → (𝜒𝜏𝜁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079
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 208  df-an 397  df-3an 1081
This theorem is referenced by:  pofun  5484  isopolem  7087  issmo2  7975  smores  7978  inawina  10100  gchina  10109  repswcshw  14162  coprmprod  15993  issubmnd  17926  issubg2  18232  issubrg2  19484  issubassa3  20025  ocv2ss  20745  sslm  21835  cmetcaulem  23818  axcontlem4  26680  axcontlem8  26684  redwlk  27381  clwwlknwwlksn  27743  numclwwlk1lem2foa  28060  dipsubdir  28552  subgrpth  32278  cgr3tr4  33410  idinside  33442  ftc1anclem7  34854  fzmul  34897  fdc1  34902  rngosubdi  35104  rngosubdir  35105  cdlemg33a  37722  upwlkwlk  43891  lidlmsgrp  44125  lidlrng  44126
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