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Mirrors > Home > MPE Home > Th. List > sb8eulem | Structured version Visualization version GIF version |
Description: Lemma. Factor out the common proof skeleton of sb8euv 2687 and sb8eu 2688. Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.) Factor out common proof lines. (Revised by Wolf Lammen, 9-Feb-2023.) |
Ref | Expression |
---|---|
sb8eulem.nfsb | ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑 |
sb8eulem.sbequ | ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
Ref | Expression |
---|---|
sb8eulem | ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 2013 | . . . . 5 ⊢ Ⅎ𝑤(𝜑 ↔ 𝑥 = 𝑧) | |
2 | 1 | sb8v 2376 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)) |
3 | equsb3v 2347 | . . . . . 6 ⊢ ([𝑤 / 𝑥]𝑥 = 𝑧 ↔ 𝑤 = 𝑧) | |
4 | 3 | sblbisv 2342 | . . . . 5 ⊢ ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ 𝑤 = 𝑧)) |
5 | 4 | albii 1918 | . . . 4 ⊢ (∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑤([𝑤 / 𝑥]𝜑 ↔ 𝑤 = 𝑧)) |
6 | sb8eulem.nfsb | . . . . . 6 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑 | |
7 | nfv 2013 | . . . . . 6 ⊢ Ⅎ𝑦 𝑤 = 𝑧 | |
8 | 6, 7 | nfbi 2006 | . . . . 5 ⊢ Ⅎ𝑦([𝑤 / 𝑥]𝜑 ↔ 𝑤 = 𝑧) |
9 | nfv 2013 | . . . . 5 ⊢ Ⅎ𝑤([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧) | |
10 | sb8eulem.sbequ | . . . . . 6 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
11 | equequ1 2129 | . . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝑧 ↔ 𝑦 = 𝑧)) | |
12 | 10, 11 | bibi12d 337 | . . . . 5 ⊢ (𝑤 = 𝑦 → (([𝑤 / 𝑥]𝜑 ↔ 𝑤 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))) |
13 | 8, 9, 12 | cbvalv1 2367 | . . . 4 ⊢ (∀𝑤([𝑤 / 𝑥]𝜑 ↔ 𝑤 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)) |
14 | 2, 5, 13 | 3bitri 289 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)) |
15 | 14 | exbii 1947 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)) |
16 | eu6 2645 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
17 | eu6 2645 | . 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑧∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)) | |
18 | 15, 16, 17 | 3bitr4i 295 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1654 ∃wex 1878 Ⅎwnf 1882 [wsb 2067 ∃!weu 2639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-10 2192 ax-11 2207 ax-12 2220 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 |
This theorem is referenced by: sb8euv 2687 sb8eu 2688 |
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