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 3200. (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 2971 | . . 3 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
3 | 2 | 19.21 2203 | . 2 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑))) |
4 | 3 | r2allem 3200 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 ∈ wcel 2110 Ⅎwnfc 2961 ∀wral 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-10 2141 ax-11 2157 ax-12 2173 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-nf 1781 df-sb 2066 df-clel 2893 df-nfc 2963 df-ral 3143 |
This theorem is referenced by: r2exf 3325 ralcomf 3357 |
Copyright terms: Public domain | W3C validator |