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| Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-ax11-lem6 | Structured version Visualization version GIF version | ||
| Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.) | 
| Ref | Expression | 
|---|---|
| wl-ax11-lem6 | ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax-wl-11v 37585 | . . 3 ⊢ (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 → ∀𝑥∀𝑢[𝑢 / 𝑦]𝜑) | |
| 2 | ax-wl-11v 37585 | . . 3 ⊢ (∀𝑥∀𝑢[𝑢 / 𝑦]𝜑 → ∀𝑢∀𝑥[𝑢 / 𝑦]𝜑) | |
| 3 | 1, 2 | impbii 209 | . 2 ⊢ (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑢[𝑢 / 𝑦]𝜑) | 
| 4 | nfna1 2152 | . . . . 5 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦 | |
| 5 | wl-ax11-lem3 37588 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑢 𝑢 = 𝑦) | |
| 6 | 4, 5 | nfan1 2200 | . . . 4 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦) | 
| 7 | wl-ax11-lem5 37590 | . . . . 5 ⊢ (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑)) | |
| 8 | 7 | adantl 481 | . . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦) → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑)) | 
| 9 | 6, 8 | albid 2222 | . . 3 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦) → (∀𝑥∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦𝜑)) | 
| 10 | 9 | ancoms 458 | . 2 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑥∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦𝜑)) | 
| 11 | 3, 10 | bitrid 283 | 1 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 [wsb 2064 | 
| 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-10 2141 ax-12 2177 ax-13 2377 ax-wl-11v 37585 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 | 
| This theorem is referenced by: wl-ax11-lem10 37595 | 
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