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Mirrors > Home > ILE Home > Th. List > ralcomf | GIF version |
Description: Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
ralcomf.1 | ⊢ Ⅎ𝑦𝐴 |
ralcomf.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
ralcomf | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancomsimp 1381 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝜑)) | |
2 | 1 | 2albii 1412 | . . 3 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) ↔ ∀𝑥∀𝑦((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝜑)) |
3 | alcom 1419 | . . 3 ⊢ (∀𝑥∀𝑦((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝜑) ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝜑)) | |
4 | 2, 3 | bitri 183 | . 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝜑)) |
5 | ralcomf.1 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
6 | 5 | r2alf 2406 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
7 | ralcomf.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
8 | 7 | r2alf 2406 | . 2 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝜑)) |
9 | 4, 6, 8 | 3bitr4i 211 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1294 ∈ wcel 1445 Ⅎwnfc 2222 ∀wral 2370 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-nf 1402 df-sb 1700 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 |
This theorem is referenced by: ralcom 2544 ssiinf 3801 |
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