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| Mirrors > Home > MPE Home > Th. List > alrimdd | Structured version Visualization version GIF version | ||
| Description: Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2207. (Contributed by Mario Carneiro, 24-Sep-2016.) | 
| Ref | Expression | 
|---|---|
| alrimdd.1 | ⊢ Ⅎ𝑥𝜑 | 
| alrimdd.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) | 
| alrimdd.3 | ⊢ (𝜑 → (𝜓 → 𝜒)) | 
| Ref | Expression | 
|---|---|
| alrimdd | ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | alrimdd.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 2 | 1 | nf5rd 2196 | . 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | 
| 3 | alrimdd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 4 | alrimdd.3 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
| 5 | 3, 4 | alimd 2212 | . 2 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) | 
| 6 | 2, 5 | syld 47 | 1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 Ⅎwnf 1783 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-nf 1784 | 
| This theorem is referenced by: alrimd 2215 wl-euequf 37575 | 
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