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Mirrors > Home > ILE Home > Th. List > r2al | GIF version |
Description: Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.) |
Ref | Expression |
---|---|
r2al | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2228 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | 1 | r2alf 2395 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1287 ∈ wcel 1438 ∀wral 2359 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 |
This theorem is referenced by: r3al 2420 raliunxp 4577 codir 4820 qfto 4821 fununi 5082 dff13 5547 mpt22eqb 5754 qliftfun 6374 |
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