Theorem 3anim123d 1329

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 335	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜒 ∧ 𝜏)))
4		3anim123d.3	. . 3 ⊢ (𝜑 → (𝜂 → 𝜁))
5	3, 4	anim12d 335	. 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜃) ∧ 𝜂) → ((𝜒 ∧ 𝜏) ∧ 𝜁)))
6		df-3an 981	. 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ ((𝜓 ∧ 𝜃) ∧ 𝜂))
7		df-3an 981	. 2 ⊢ ((𝜒 ∧ 𝜏 ∧ 𝜁) ↔ ((𝜒 ∧ 𝜏) ∧ 𝜁))
8	5, 6, 7	3imtr4g 205	1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) → (𝜒 ∧ 𝜏 ∧ 𝜁)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 981

This theorem is referenced by:  hb3and  1500  pofun  4324  soss  4326  wessep  4589  isopolem  5836  isosolem  5838  issmo2  6303  smores  6306  issubmnd  12862  issubg2m  13078  issubrg2  13425  sslm  13987