Theorem 3anim123d 1445

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 609	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜒 ∧ 𝜏)))
4		3anim123d.3	. . 3 ⊢ (𝜑 → (𝜂 → 𝜁))
5	3, 4	anim12d 609	. 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜃) ∧ 𝜂) → ((𝜒 ∧ 𝜏) ∧ 𝜁)))
6		df-3an 1088	. 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ ((𝜓 ∧ 𝜃) ∧ 𝜂))
7		df-3an 1088	. 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 1086

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 1088

This theorem is referenced by:  pofun  5567  isopolem  7323  issmo2  8321  smores  8324  inawina  10650  gchina  10659  repswcshw  14784  coprmprod  16638  issubmnd  18695  issubg2  19080  issubrng2  20474  issubrg2  20508  rnglidlmsgrp  21163  rnglidlrng  21164  ocv2ss  21589  issubassa3  21782  sslm  23193  cmetcaulem  25195  axcontlem4  28901  axcontlem8  28905  redwlk  29607  clwwlknwwlksn  29974  numclwwlk1lem2foa  30290  dipsubdir  30784  constrconj  33742  subgrpth  35128  cgr3tr4  36047  idinside  36079  ftc1anclem7  37700  fzmul  37742  fdc1  37747  rngosubdi  37946  rngosubdir  37947  cdlemg33a  40707  grtrimap  47951  grimgrtri  47952  grlimgrtri  47999  upwlkwlk  48131