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Mirrors > Home > ILE Home > Th. List > 3orim123d | GIF version |
Description: Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.) |
Ref | Expression |
---|---|
3anim123d.1 | ⊢ (φ → (ψ → χ)) |
3anim123d.2 | ⊢ (φ → (θ → τ)) |
3anim123d.3 | ⊢ (φ → (η → ζ)) |
Ref | Expression |
---|---|
3orim123d | ⊢ (φ → ((ψ ∨ θ ∨ η) → (χ ∨ τ ∨ ζ))) |
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))) |
Colors of variables: wff set class |
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 |
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