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Mathbox for Wolf Lammen |
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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 33678. (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 2185 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑤 | |
2 | nfeqf2 2452 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑤 → Ⅎ𝑥 𝑣 = 𝑤) | |
3 | 1, 2 | nfan1 2222 | . . . . 5 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑤 ∧ 𝑣 = 𝑤) |
4 | sbequ 2523 | . . . . . 6 ⊢ (𝑣 = 𝑤 → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑)) | |
5 | 4 | adantl 467 | . . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑤 ∧ 𝑣 = 𝑤) → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑)) |
6 | 3, 5 | sbbid 2550 | . . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑤 ∧ 𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑)) |
7 | 6 | ancoms 455 | . . 3 ⊢ ((𝑣 = 𝑤 ∧ ¬ ∀𝑥 𝑥 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑)) |
8 | sbequ 2523 | . . 3 ⊢ (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)) | |
9 | 7, 8 | sylan9bbr 500 | . 2 ⊢ ((𝑢 = 𝑦 ∧ (𝑣 = 𝑤 ∧ ¬ ∀𝑥 𝑥 = 𝑤)) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)) |
10 | 9 | expr 444 | 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 196 ∧ wa 382 ∀wal 1629 [wsb 2049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-10 2174 ax-12 2203 ax-13 2408 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-ex 1853 df-nf 1858 df-sb 2050 |
This theorem is referenced by: wl-sbcom2d 33678 |
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