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 → (𝜓 → 𝜒))
3anim123d.2	⊢ (𝜑 → (𝜃 → 𝜏))
3anim123d.3	⊢ (𝜑 → (𝜂 → 𝜁))

Assertion

Ref	Expression
3anim123d	⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) → (𝜒 ∧ 𝜏 ∧ 𝜁)))

Proof of Theorem 3anim123d

Step	Hyp	Ref	Expression
1		3anim123d.1	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
2		3anim123d.2	. . . 4 ⊢ (𝜑 → (𝜃 → 𝜏))
3	1, 2	anim12d 610	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜒 ∧ 𝜏)))
4		3anim123d.3	. . 3 ⊢ (𝜑 → (𝜂 → 𝜁))
5	3, 4	anim12d 610	. 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜃) ∧ 𝜂) → ((𝜒 ∧ 𝜏) ∧ 𝜁)))
6		df-3an 1089	. 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ ((𝜓 ∧ 𝜃) ∧ 𝜂))
7		df-3an 1089	. 2 ⊢ ((𝜒 ∧ 𝜏 ∧ 𝜁) ↔ ((𝜒 ∧ 𝜏) ∧ 𝜁))
8	5, 6, 7	3imtr4g 296	1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) → (𝜒 ∧ 𝜏 ∧ 𝜁)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087

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-3an 1089

This theorem is referenced by:  pofun  5558  isopolem  7301  issmo2  8291  smores  8294  inawina  10613  gchina  10622  repswcshw  14747  coprmprod  16600  issubmnd  18698  issubg2  19083  issubrng2  20503  issubrg2  20537  rnglidlmsgrp  21213  rnglidlrng  21214  ocv2ss  21640  issubassa3  21833  sslm  23255  cmetcaulem  25256  bdayfinbndlem1  28475  axcontlem4  29052  axcontlem8  29056  redwlk  29756  clwwlknwwlksn  30125  numclwwlk1lem2foa  30441  dipsubdir  30935  constrconj  33922  subgrpth  35347  cgr3tr4  36265  idinside  36297  ftc1anclem7  37947  fzmul  37989  fdc1  37994  rngosubdi  38193  rngosubdir  38194  cdlemg33a  41079  grtrimap  48305  grimgrtri  48306  grlimgrtri  48360  upwlkwlk  48496