Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 2sb8e | Structured version Visualization version GIF version |
Description: An equivalent expression for double existence. Usage of this theorem is discouraged because it depends on ax-13 2389. For a version requiring more disjoint variables, but fewer axioms, see 2sb8ev 2374. (Contributed by Wolf Lammen, 2-Nov-2019.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2sb8e | ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑤𝜑 | |
2 | 1 | sb8e 2559 | . . . 4 ⊢ (∃𝑦𝜑 ↔ ∃𝑤[𝑤 / 𝑦]𝜑) |
3 | 2 | exbii 1847 | . . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑤[𝑤 / 𝑦]𝜑) |
4 | excom 2168 | . . 3 ⊢ (∃𝑥∃𝑤[𝑤 / 𝑦]𝜑 ↔ ∃𝑤∃𝑥[𝑤 / 𝑦]𝜑) | |
5 | 3, 4 | bitri 277 | . 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑤∃𝑥[𝑤 / 𝑦]𝜑) |
6 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑧𝜑 | |
7 | 6 | nfsb 2564 | . . . 4 ⊢ Ⅎ𝑧[𝑤 / 𝑦]𝜑 |
8 | 7 | sb8e 2559 | . . 3 ⊢ (∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
9 | 8 | exbii 1847 | . 2 ⊢ (∃𝑤∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑤∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
10 | excom 2168 | . 2 ⊢ (∃𝑤∃𝑧[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) | |
11 | 5, 9, 10 | 3bitri 299 | 1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∃wex 1779 [wsb 2068 |
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 1969 ax-7 2014 ax-10 2144 ax-11 2160 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |