Theorem 3orim123d 1260

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 811	. . 3 ⊢ (φ → ((ψ ∨ θ) → (χ ∨ τ)))
4		3anim123d.3	. . 3 ⊢ (φ → (η → ζ))
5	3, 4	orim12d 811	. 2 ⊢ (φ → (((ψ ∨ θ) ∨ η) → ((χ ∨ τ) ∨ ζ)))
6		df-3or 935	. 2 ⊢ ((ψ ∨ θ ∨ η) ↔ ((ψ ∨ θ) ∨ η))
7		df-3or 935	. 2 ⊢ ((χ ∨ τ ∨ ζ) ↔ ((χ ∨ τ) ∨ ζ))
8	5, 6, 7	3imtr4g 261	1 ⊢ (φ → ((ψ ∨ θ ∨ η) → (χ ∨ τ ∨ ζ)))

This proof depends on syntax axioms:   → wi 4   ∨ wo 357   ∨ w3o 933

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935

This theorem is used by:  nncdiv3  6278