|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > r2alf | Structured version Visualization version GIF version | ||
| Description: Double restricted universal quantification. For a version based on fewer axioms see r2al 3195. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2allem 3142. (Revised by Wolf Lammen, 9-Jan-2020.) | 
| Ref | Expression | 
|---|---|
| r2alf.1 | ⊢ Ⅎ𝑦𝐴 | 
| Ref | Expression | 
|---|---|
| r2alf | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | r2alf.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
| 2 | 1 | nfcri 2897 | . . 3 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | 
| 3 | 2 | 19.21 2207 | . 2 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑))) | 
| 4 | 3 | r2allem 3142 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 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-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: r2exf 3282 ralcomf 3302 | 
| Copyright terms: Public domain | W3C validator |