Theorem 3orim123d 1446

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 966	. . 3 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → (𝜒 ∨ 𝜏)))
4		3anim123d.3	. . 3 ⊢ (𝜑 → (𝜂 → 𝜁))
5	3, 4	orim12d 966	. 2 ⊢ (𝜑 → (((𝜓 ∨ 𝜃) ∨ 𝜂) → ((𝜒 ∨ 𝜏) ∨ 𝜁)))
6		df-3or 1087	. 2 ⊢ ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂))
7		df-3or 1087	. 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 847   ∨ w3o 1085

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 848  df-3or 1087

This theorem is referenced by:  fr3nr  7717  soxp  8071  poxp3  8092  zorn2lem6  10411  fpwwe2lem11  10552  fpwwe2lem12  10553  chnso  18547  ltsres  27630  colinearalglem4  28982  constrconj  33902  vonf1owev  35302  colinearxfr  36269  weiunso  36660  fin2so  37808  frege133d  44006  chnerlem3  47128  el1fzopredsuc  47571  fmtno4prmfac  47818