Theorem 3orim123d 1214

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 699	. . 3 ⊢ (φ → ((ψ ∨ θ) → (χ ∨ τ)))
4		3anim123d.3	. . 3 ⊢ (φ → (η → ζ))
5	3, 4	orim12d 699	. 2 ⊢ (φ → (((ψ ∨ θ) ∨ η) → ((χ ∨ τ) ∨ ζ)))
6		df-3or 885	. 2 ⊢ ((ψ ∨ θ ∨ η) ↔ ((ψ ∨ θ) ∨ η))
7		df-3or 885	. 2 ⊢ ((χ ∨ τ ∨ ζ) ↔ ((χ ∨ τ) ∨ ζ))
8	5, 6, 7	3imtr4g 194	1 ⊢ (φ → ((ψ ∨ θ ∨ η) → (χ ∨ τ ∨ ζ)))

Syntax hints:   → wi 4   ∨ wo 628   ∨ w3o 883

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629

This theorem depends on definitions:  df-bi 110  df-3or 885

This theorem is referenced by:  ztri3or0  8063