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| Mirrors > Home > MPE Home > Th. List > eean | Structured version Visualization version GIF version | ||
| Description: Distribute existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.) |
| Ref | Expression |
|---|---|
| eean.1 | ⊢ Ⅎ𝑦𝜑 |
| eean.2 | ⊢ Ⅎ𝑥𝜓 |
| Ref | Expression |
|---|---|
| eean | ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eean.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 2 | 1 | 19.42 2236 | . . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓)) |
| 3 | 2 | exbii 1848 | . 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) |
| 4 | eean.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 5 | 4 | nfex 2324 | . . 3 ⊢ Ⅎ𝑥∃𝑦𝜓 |
| 6 | 5 | 19.41 2235 | . 2 ⊢ (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
| 7 | 3, 6 | bitri 275 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1779 Ⅎwnf 1783 |
| 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 |
| This theorem is referenced by: eeanv 2351 ee4anv 2353 reean 3316 |
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