Theorem 3orim123d 1472

Description: Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.)

Hypotheses

Ref	Expression
3anim123d.1	⊢ (𝜑 → (𝜓 → 𝜒))
3anim123d.2	⊢ (𝜑 → (𝜃 → 𝜏))
3anim123d.3	⊢ (𝜑 → (𝜂 → 𝜁))

Assertion

Ref	Expression
3orim123d	⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜂) → (𝜒 ∨ 𝜏 ∨ 𝜁)))

Proof of Theorem 3orim123d

Step	Hyp	Ref	Expression
1		3anim123d.1	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
2		3anim123d.2	. . . 4 ⊢ (𝜑 → (𝜃 → 𝜏))
3	1, 2	orim12d 979	. . 3 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → (𝜒 ∨ 𝜏)))
4		3anim123d.3	. . 3 ⊢ (𝜑 → (𝜂 → 𝜁))
5	3, 4	orim12d 979	. 2 ⊢ (𝜑 → (((𝜓 ∨ 𝜃) ∨ 𝜂) → ((𝜒 ∨ 𝜏) ∨ 𝜁)))
6		df-3or 1104	. 2 ⊢ ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂))
7		df-3or 1104	. 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   ∨ wo 860   ∨ w3o 1102

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-or 861  df-3or 1104

This theorem is referenced by:  3orim123da  1473  fr3nr  7767  soxp  8121  poxp3  8142  zorn2lem6  10480  fpwwe2lem11  10621  fpwwe2lem12  10622  chnso  18675  ltsres  27826  colinearalglem4  29259  constrconj  34135  vonf1wev  35592  vonf1owevOLD  35594  colinearxfr  36567  weiunso  36977  fin2so  38258  frege133d  44491  chnerlem3  47600  el1fzopredsuc  48063  fmtno4prmfac  48324