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Mirrors > Home > ILE Home > Th. List > rexcomf | GIF version |
Description: Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
ralcomf.1 | ⊢ Ⅎ𝑦𝐴 |
ralcomf.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
rexcomf | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 262 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
2 | 1 | anbi1i 446 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑)) |
3 | 2 | 2exbii 1542 | . . 3 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑)) |
4 | excom 1599 | . . 3 ⊢ (∃𝑥∃𝑦((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑)) | |
5 | 3, 4 | bitri 182 | . 2 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑)) |
6 | ralcomf.1 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
7 | 6 | r2exf 2396 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) |
8 | ralcomf.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
9 | 8 | r2exf 2396 | . 2 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑)) |
10 | 5, 7, 9 | 3bitr4i 210 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∃wex 1426 ∈ wcel 1438 Ⅎwnfc 2215 ∃wrex 2360 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 df-cleq 2081 df-clel 2084 df-nfc 2217 df-rex 2365 |
This theorem is referenced by: rexcom 2531 |
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