Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 2exsb | Structured version Visualization version GIF version |
Description: An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 30-Sep-2018.) |
Ref | Expression |
---|---|
2exsb | ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1916 | . . 3 ⊢ Ⅎ𝑤𝜑 | |
2 | nfv 1916 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
3 | 1, 2 | 2sb8ef 2352 | . 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) |
4 | 2sb6 2088 | . . 3 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑)) | |
5 | 4 | 2exbii 1850 | . 2 ⊢ (∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑)) |
6 | 3, 5 | bitri 274 | 1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1538 ∃wex 1780 [wsb 2066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-10 2136 ax-11 2153 ax-12 2170 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1781 df-nf 1785 df-sb 2067 |
This theorem is referenced by: 2eu6 2656 |
Copyright terms: Public domain | W3C validator |