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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-equsb3 | Structured version Visualization version GIF version |
Description: equsb3 2420 with a distinctor. (Contributed by Wolf Lammen, 27-Jun-2019.) |
Ref | Expression |
---|---|
wl-equsb3 | ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . . 3 ⊢ Ⅎ𝑤 ¬ ∀𝑦 𝑦 = 𝑧 | |
2 | nfna1 2016 | . . . 4 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑧 | |
3 | nfeqf2 2285 | . . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤 = 𝑧) | |
4 | equequ1 1939 | . . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 = 𝑧 ↔ 𝑤 = 𝑧)) | |
5 | 4 | a1i 11 | . . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (𝑦 = 𝑤 → (𝑦 = 𝑧 ↔ 𝑤 = 𝑧))) |
6 | 2, 3, 5 | sbied 2397 | . . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑤 / 𝑦]𝑦 = 𝑧 ↔ 𝑤 = 𝑧)) |
7 | 1, 6 | sbbid 2391 | . 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧)) |
8 | sbcom3 2399 | . . 3 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤][𝑥 / 𝑦]𝑦 = 𝑧) | |
9 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑤[𝑥 / 𝑦]𝑦 = 𝑧 | |
10 | 9 | sbf 2368 | . . 3 ⊢ ([𝑥 / 𝑤][𝑥 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧) |
11 | 8, 10 | bitri 263 | . 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧) |
12 | equsb3 2420 | . 2 ⊢ ([𝑥 / 𝑤]𝑤 = 𝑧 ↔ 𝑥 = 𝑧) | |
13 | 7, 11, 12 | 3bitr3g 301 | 1 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∀wal 1473 [wsb 1867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 |
This theorem is referenced by: (None) |
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