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| Mirrors > Home > MPE Home > Th. List > ralcomf | Structured version Visualization version GIF version | ||
| Description: Commutation of restricted universal quantifiers. For a version based on fewer axioms see ralcom 3289. (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 | ancomst 464 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝜑)) | |
| 2 | 1 | 2albii 1820 | . . 3 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) ↔ ∀𝑥∀𝑦((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝜑)) |
| 3 | alcom 2159 | . . 3 ⊢ (∀𝑥∀𝑦((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝜑) ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝜑)) | |
| 4 | 2, 3 | bitri 275 | . 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝜑)) |
| 5 | ralcomf.1 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
| 6 | 5 | r2alf 3281 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
| 7 | ralcomf.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
| 8 | 7 | r2alf 3281 | . 2 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝜑)) |
| 9 | 4, 6, 8 | 3bitr4i 303 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 |
| 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-8 2110 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-clel 2816 df-nfc 2892 df-ral 3062 |
| This theorem is referenced by: ssiinf 5054 ralcom4f 32486 |
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