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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 2332 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | 1 | r2alf 2507 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1362 ∈ wcel 2160 ∀wral 2468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 |
This theorem is referenced by: r3al 2534 raliunxp 4786 codir 5035 qfto 5036 fununi 5303 dff13 5790 mpo2eqb 6006 qliftfun 6643 cnmpt12 14247 cnmpt22 14254 |
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