Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > reean | GIF version |
Description: Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
reean.1 | ⊢ Ⅎ𝑦𝜑 |
reean.2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
reean | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | an4 520 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝜑 ∧ 𝜓)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) | |
2 | 1 | 2exbii 1497 | . . 3 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝜑 ∧ 𝜓)) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) |
3 | nfv 1421 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
4 | reean.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
5 | 3, 4 | nfan 1457 | . . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) |
6 | nfv 1421 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 | |
7 | reean.2 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
8 | 6, 7 | nfan 1457 | . . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐵 ∧ 𝜓) |
9 | 5, 8 | eean 1806 | . . 3 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐵 ∧ 𝜓)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜓))) |
10 | 2, 9 | bitri 173 | . 2 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜓))) |
11 | r2ex 2344 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝜑 ∧ 𝜓))) | |
12 | df-rex 2312 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
13 | df-rex 2312 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜓)) | |
14 | 12, 13 | anbi12i 433 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜓))) |
15 | 10, 11, 14 | 3bitr4i 201 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 Ⅎwnf 1349 ∃wex 1381 ∈ wcel 1393 ∃wrex 2307 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 |
This theorem is referenced by: reeanv 2479 |
Copyright terms: Public domain | W3C validator |