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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 2530 with more disjoint variable conditions, not requiring ax-13 2372. (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 2352 | . . . 4 ⊢ (∃𝑦𝜑 ↔ ∃𝑤[𝑤 / 𝑦]𝜑) |
3 | 2 | exbii 1851 | . . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑤[𝑤 / 𝑦]𝜑) |
4 | excom 2163 | . . 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 2324 | . . . 4 ⊢ Ⅎ𝑧[𝑤 / 𝑦]𝜑 |
8 | 7 | sb8ef 2352 | . . 3 ⊢ (∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
9 | 8 | exbii 1851 | . 2 ⊢ (∃𝑤∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑤∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
10 | excom 2163 | . 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 205 ∃wex 1782 Ⅎwnf 1786 [wsb 2068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-11 2155 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ex 1783 df-nf 1787 df-sb 2069 |
This theorem is referenced by: 2exsb 2357 2mo 2645 |
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