![]() |
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 3188. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2allem 3136. (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 2891 | . . 3 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
3 | 2 | 19.21 2201 | . 2 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑))) |
4 | 3 | r2allem 3136 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 ∈ wcel 2107 Ⅎwnfc 2884 ∀wral 3061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 df-clel 2811 df-nfc 2886 df-ral 3062 |
This theorem is referenced by: r2exf 3264 ralcomf 3284 |
Copyright terms: Public domain | W3C validator |