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Mirrors > Home > NFE Home > Th. List > syl3anbrc | GIF version |
Description: Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.) |
Ref | Expression |
---|---|
syl3anbrc.1 | ⊢ (φ → ψ) |
syl3anbrc.2 | ⊢ (φ → χ) |
syl3anbrc.3 | ⊢ (φ → θ) |
syl3anbrc.4 | ⊢ (τ ↔ (ψ ∧ χ ∧ θ)) |
Ref | Expression |
---|---|
syl3anbrc | ⊢ (φ → τ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3anbrc.1 | . . 3 ⊢ (φ → ψ) | |
2 | syl3anbrc.2 | . . 3 ⊢ (φ → χ) | |
3 | syl3anbrc.3 | . . 3 ⊢ (φ → θ) | |
4 | 1, 2, 3 | 3jca 1132 | . 2 ⊢ (φ → (ψ ∧ χ ∧ θ)) |
5 | syl3anbrc.4 | . 2 ⊢ (τ ↔ (ψ ∧ χ ∧ θ)) | |
6 | 4, 5 | sylibr 203 | 1 ⊢ (φ → τ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ 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: sfindbl 4530 vfinspsslem1 4550 pod 5936 fundmen 6043 enmap1 6074 enprmap 6082 |
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