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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-sbcom2d-lem1 | Structured version Visualization version GIF version |
Description: Lemma used to prove wl-sbcom2d 34679. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.) |
Ref | Expression |
---|---|
wl-sbcom2d-lem1 | ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfna1 2147 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑤 | |
2 | nfeqf2 2388 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑤 → Ⅎ𝑥 𝑣 = 𝑤) | |
3 | 1, 2 | nfan1 2191 | . . . . 5 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑤 ∧ 𝑣 = 𝑤) |
4 | sbequ 2081 | . . . . . 6 ⊢ (𝑣 = 𝑤 → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑)) | |
5 | 4 | adantl 482 | . . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑤 ∧ 𝑣 = 𝑤) → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑)) |
6 | 3, 5 | sbbid 2237 | . . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑤 ∧ 𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑)) |
7 | 6 | ancoms 459 | . . 3 ⊢ ((𝑣 = 𝑤 ∧ ¬ ∀𝑥 𝑥 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑)) |
8 | sbequ 2081 | . . 3 ⊢ (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)) | |
9 | 7, 8 | sylan9bbr 511 | . 2 ⊢ ((𝑢 = 𝑦 ∧ (𝑣 = 𝑤 ∧ ¬ ∀𝑥 𝑥 = 𝑤)) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)) |
10 | 9 | expr 457 | 1 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 ax-13 2383 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 |
This theorem is referenced by: wl-sbcom2d 34679 |
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