Theorem im3ord 637

Description: Deduction joining 3 implications to form implication of disjunctions.

Hypotheses

Ref	Expression
im3d.1	⊢ (φ → (ψ → χ))
im3d.2	⊢ (φ → (θ → τ))
im3d.3	⊢ (φ → (η → ζ))

Assertion

Ref	Expression
im3ord	⊢ (φ → ((ψ ∨ θ ∨ η) → (χ ∨ τ ∨ ζ)))

Proof of Theorem im3ord

Step	Hyp	Ref	Expression
1		im3d.1	. . . 4 ⊢ (φ → (ψ → χ))
2		im3d.2	. . . 4 ⊢ (φ → (θ → τ))
3	1, 2	orim12d 436	. . 3 ⊢ (φ → ((ψ ∨ θ) → (χ ∨ τ)))
4		im3d.3	. . 3 ⊢ (φ → (η → ζ))
5	3, 4	orim12d 436	. 2 ⊢ (φ → (((ψ ∨ θ) ∨ η) → ((χ ∨ τ) ∨ ζ)))
6		df-3or 582	. 2 ⊢ ((ψ ∨ θ ∨ η) ↔ ((ψ ∨ θ) ∨ η))
7		df-3or 582	. 2 ⊢ ((χ ∨ τ ∨ ζ) ↔ ((χ ∨ τ) ∨ ζ))
8	5, 6, 7	3imtr4g 426	1 ⊢ (φ → ((ψ ∨ θ ∨ η) → (χ ∨ τ ∨ ζ)))

Syntax hints:   → wi 2   ∨ wo 195   ∨ w3o 580

This theorem is referenced by:  zornlem6 3608

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582

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