Theorem 3orim123d 1444

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 965	. . 3 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → (𝜒 ∨ 𝜏)))
4		3anim123d.3	. . 3 ⊢ (𝜑 → (𝜂 → 𝜁))
5	3, 4	orim12d 965	. 2 ⊢ (𝜑 → (((𝜓 ∨ 𝜃) ∨ 𝜂) → ((𝜒 ∨ 𝜏) ∨ 𝜁)))
6		df-3or 1088	. 2 ⊢ ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂))
7		df-3or 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   ∨ wo 846   ∨ w3o 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-or 847  df-3or 1088

This theorem is referenced by:  fr3nr  7807  soxp  8170  poxp3  8191  zorn2lem6  10570  fpwwe2lem11  10710  fpwwe2lem12  10711  sltres  27725  colinearalglem4  28942  chnso  32986  constrconj  33735  colinearxfr  36039  weiunso  36432  fin2so  37567  frege133d  43727  el1fzopredsuc  47240  fmtno4prmfac  47446