Theorem 3anim123d 1471

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 620	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜒 ∧ 𝜏)))
4		3anim123d.3	. . 3 ⊢ (𝜑 → (𝜂 → 𝜁))
5	3, 4	anim12d 620	. 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜃) ∧ 𝜂) → ((𝜒 ∧ 𝜏) ∧ 𝜁)))
6		df-3an 1105	. 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ ((𝜓 ∧ 𝜃) ∧ 𝜂))
7		df-3an 1105	. 2 ⊢ ((𝜒 ∧ 𝜏 ∧ 𝜁) ↔ ((𝜒 ∧ 𝜏) ∧ 𝜁))
8	5, 6, 7	3imtr4g 299	1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) → (𝜒 ∧ 𝜏 ∧ 𝜁)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103

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 210  df-an 401  df-3an 1105

This theorem is referenced by:  pofun  5589  isopolem  7345  issmo2  8337  smores  8340  inawina  10676  gchina  10685  repswcshw  14851  coprmprod  16720  issubmnd  18820  issubg2  19209  issubrng2  20644  issubrg2  20678  rnglidlmsgrp  21361  rnglidlrng  21362  ocv2ss  21804  issubassa3  21997  sslm  23437  cmetcaulem  25428  bdayfinbndlem1  28641  axcontlem4  29298  axcontlem8  29302  redwlk  30001  clwwlknwwlksn  30370  numclwwlk1lem2foa  30686  dipsubdir  31181  constrconj  34116  subgrpth  35607  cgr3tr4  36525  idinside  36557  ftc1anclem7  38331  fzmul  38373  fdc1  38378  rngosubdi  38577  rngosubdir  38578  cdlemg33a  41461  grtrimap  48696  grimgrtri  48697  grlimgrtri  48751  upwlkwlk  48887