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Mirrors > Home > NFE Home > Th. List > syl3an9b | GIF version |
Description: Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.) |
Ref | Expression |
---|---|
syl3an9b.1 | ⊢ (φ → (ψ ↔ χ)) |
syl3an9b.2 | ⊢ (θ → (χ ↔ τ)) |
syl3an9b.3 | ⊢ (η → (τ ↔ ζ)) |
Ref | Expression |
---|---|
syl3an9b | ⊢ ((φ ∧ θ ∧ η) → (ψ ↔ ζ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3an9b.1 | . . . 4 ⊢ (φ → (ψ ↔ χ)) | |
2 | syl3an9b.2 | . . . 4 ⊢ (θ → (χ ↔ τ)) | |
3 | 1, 2 | sylan9bb 680 | . . 3 ⊢ ((φ ∧ θ) → (ψ ↔ τ)) |
4 | syl3an9b.3 | . . 3 ⊢ (η → (τ ↔ ζ)) | |
5 | 3, 4 | sylan9bb 680 | . 2 ⊢ (((φ ∧ θ) ∧ η) → (ψ ↔ ζ)) |
6 | 5 | 3impa 1146 | 1 ⊢ ((φ ∧ θ ∧ η) → (ψ ↔ ζ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∧ w3a 934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 177 df-an 360 df-3an 936 |
This theorem is referenced by: eloprabg 5579 |
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