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Mirrors > Home > ILE Home > Th. List > raaanv | GIF version |
Description: Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.) |
Ref | Expression |
---|---|
raaanv | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1493 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1493 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | 1, 2 | raaan 3439 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∀wral 2393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 |
This theorem is referenced by: reusv3i 4350 f1mpt 5640 |
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