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Mirrors > Home > ILE Home > Th. List > 3orbi123i | GIF version |
Description: Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.) |
Ref | Expression |
---|---|
bi3.1 | ⊢ (φ ↔ ψ) |
bi3.2 | ⊢ (χ ↔ θ) |
bi3.3 | ⊢ (τ ↔ η) |
Ref | Expression |
---|---|
3orbi123i | ⊢ ((φ ∨ χ ∨ τ) ↔ (ψ ∨ θ ∨ η)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi3.1 | . . . 4 ⊢ (φ ↔ ψ) | |
2 | bi3.2 | . . . 4 ⊢ (χ ↔ θ) | |
3 | 1, 2 | orbi12i 680 | . . 3 ⊢ ((φ ∨ χ) ↔ (ψ ∨ θ)) |
4 | bi3.3 | . . 3 ⊢ (τ ↔ η) | |
5 | 3, 4 | orbi12i 680 | . 2 ⊢ (((φ ∨ χ) ∨ τ) ↔ ((ψ ∨ θ) ∨ η)) |
6 | df-3or 885 | . 2 ⊢ ((φ ∨ χ ∨ τ) ↔ ((φ ∨ χ) ∨ τ)) | |
7 | df-3or 885 | . 2 ⊢ ((ψ ∨ θ ∨ η) ↔ ((ψ ∨ θ) ∨ η)) | |
8 | 5, 6, 7 | 3bitr4i 201 | 1 ⊢ ((φ ∨ χ ∨ τ) ↔ (ψ ∨ θ ∨ η)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∨ 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: (None) |
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