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| Mirrors > Home > MPE Home > Th. List > 2sb8ef | Structured version Visualization version GIF version | ||
| Description: An equivalent expression for double existence. Version of 2sb8e 2535 with more disjoint variable conditions, not requiring ax-13 2377. (Contributed by Wolf Lammen, 28-Jan-2023.) |
| Ref | Expression |
|---|---|
| 2sb8ef.1 | ⊢ Ⅎ𝑤𝜑 |
| 2sb8ef.2 | ⊢ Ⅎ𝑧𝜑 |
| Ref | Expression |
|---|---|
| 2sb8ef | ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 2sb8ef.1 | . . . . 5 ⊢ Ⅎ𝑤𝜑 | |
| 2 | 1 | sb8ef 2358 | . . . 4 ⊢ (∃𝑦𝜑 ↔ ∃𝑤[𝑤 / 𝑦]𝜑) |
| 3 | 2 | exbii 1848 | . . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑤[𝑤 / 𝑦]𝜑) |
| 4 | excom 2162 | . . 3 ⊢ (∃𝑥∃𝑤[𝑤 / 𝑦]𝜑 ↔ ∃𝑤∃𝑥[𝑤 / 𝑦]𝜑) | |
| 5 | 3, 4 | bitri 275 | . 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑤∃𝑥[𝑤 / 𝑦]𝜑) |
| 6 | 2sb8ef.2 | . . . . 5 ⊢ Ⅎ𝑧𝜑 | |
| 7 | 6 | nfsbv 2330 | . . . 4 ⊢ Ⅎ𝑧[𝑤 / 𝑦]𝜑 |
| 8 | 7 | sb8ef 2358 | . . 3 ⊢ (∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
| 9 | 8 | exbii 1848 | . 2 ⊢ (∃𝑤∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑤∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
| 10 | excom 2162 | . 2 ⊢ (∃𝑤∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) | |
| 11 | 5, 9, 10 | 3bitri 297 | 1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∃wex 1779 Ⅎwnf 1783 [wsb 2064 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 |
| This theorem is referenced by: 2exsb 2363 2mo 2648 |
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