Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  3anim123d Structured version   Visualization version   GIF version

Theorem 3anim123d 1446
 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 585 . . 3 (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))
4 3anim123d.3 . . 3 (𝜑 → (𝜂𝜁))
53, 4anim12d 585 . 2 (𝜑 → (((𝜓𝜃) ∧ 𝜂) → ((𝜒𝜏) ∧ 𝜁)))
6 df-3an 1056 . 2 ((𝜓𝜃𝜂) ↔ ((𝜓𝜃) ∧ 𝜂))
7 df-3an 1056 . 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 1054 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 1056 This theorem is referenced by:  pofun  5080  isopolem  6635  issmo2  7491  smores  7494  inawina  9550  gchina  9559  repswcshw  13604  coprmprod  15422  issubmnd  17365  issubg2  17656  issubrg2  18848  ocv2ss  20065  sslm  21151  cmetcaulem  23132  axcontlem4  25892  axcontlem8  25896  redwlk  26625  clwwlknwwlksn  27000  clwwlknwwlksnOLD  27001  clwwlknonwwlknonb  27080  numclwlk1lem2foa  27344  dipsubdir  27831  cgr3tr4  32284  idinside  32316  ftc1anclem7  33621  fzmul  33667  fdc1  33672  rngosubdi  33874  rngosubdir  33875  cdlemg33a  36311  upwlkwlk  42045  lidlmsgrp  42251  lidlrng  42252
 Copyright terms: Public domain W3C validator