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. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2allem 3197. (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 2968 | . . 3 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
3 | 2 | 19.21 2197 | . 2 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑))) |
4 | 3 | r2allem 3197 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 ∈ wcel 2105 Ⅎwnfc 2958 ∀wral 3135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-10 2136 ax-11 2151 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-sb 2061 df-clel 2890 df-nfc 2960 df-ral 3140 |
This theorem is referenced by: r2exf 3322 ralcomf 3354 |
Copyright terms: Public domain | W3C validator |