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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 35381 | . . 3 ⊢ (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 → ∀𝑥∀𝑢[𝑢 / 𝑦]𝜑) | |
2 | ax-wl-11v 35381 | . . 3 ⊢ (∀𝑥∀𝑢[𝑢 / 𝑦]𝜑 → ∀𝑢∀𝑥[𝑢 / 𝑦]𝜑) | |
3 | 1, 2 | impbii 212 | . 2 ⊢ (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑢[𝑢 / 𝑦]𝜑) |
4 | nfna1 2157 | . . . . 5 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦 | |
5 | wl-ax11-lem3 35384 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑢 𝑢 = 𝑦) | |
6 | 4, 5 | nfan1 2202 | . . . 4 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦) |
7 | wl-ax11-lem5 35386 | . . . . 5 ⊢ (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑)) | |
8 | 7 | adantl 485 | . . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦) → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑)) |
9 | 6, 8 | albid 2224 | . . 3 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦) → (∀𝑥∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦𝜑)) |
10 | 9 | ancoms 462 | . 2 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑥∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦𝜑)) |
11 | 3, 10 | syl5bb 286 | 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 209 ∧ wa 399 ∀wal 1540 [wsb 2074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-12 2179 ax-13 2373 ax-wl-11v 35381 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 |
This theorem is referenced by: wl-ax11-lem10 35391 |
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