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Mirrors > Home > ILE Home > Th. List > r2exf | GIF version |
Description: Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
r2alf.1 | ⊢ Ⅎ𝑦𝐴 |
Ref | Expression |
---|---|
r2exf | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2399 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑)) | |
2 | r2alf.1 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
3 | 2 | nfcri 2252 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
4 | 3 | 19.42 1651 | . . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑))) |
5 | anass 398 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑))) | |
6 | 5 | exbii 1569 | . . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑))) |
7 | df-rex 2399 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)) | |
8 | 7 | anbi2i 452 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑))) |
9 | 4, 6, 8 | 3bitr4i 211 | . . 3 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑)) |
10 | 9 | exbii 1569 | . 2 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑)) |
11 | 1, 10 | bitr4i 186 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1453 ∈ wcel 1465 Ⅎwnfc 2245 ∃wrex 2394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 |
This theorem is referenced by: r2ex 2432 rexcomf 2570 |
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