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| Mirrors > Home > MPE Home > Th. List > nfra2w | Structured version Visualization version GIF version | ||
| Description: Similar to Lemma 24 of [Monk2] p. 114, except that quantification is restricted. Once derived from hbra2VD 44885. Version of nfra2 3375 with a disjoint variable condition not requiring ax-13 2376. (Contributed by Alan Sare, 31-Dec-2011.) Reduce axiom usage. (Revised by GG, 24-Sep-2024.) (Proof shortened by Wolf Lammen, 3-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| nfra2w | ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | r2al 3194 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) | |
| 2 | nfa2 2175 | . 2 ⊢ Ⅎ𝑦∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) | |
| 3 | 1, 2 | nfxfr 1852 | 1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1537 Ⅎwnf 1782 ∈ wcel 2107 ∀wral 3060 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-10 2140 ax-11 2156 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-nf 1783 df-ral 3061 | 
| This theorem is referenced by: invdisj 5128 reusv3 5404 dedekind 11425 dedekindle 11426 mreexexd 17692 gsummatr01lem4 22665 ordtconnlem1 33924 bnj1379 34845 tratrb 44561 islptre 45639 sprsymrelfo 47489 | 
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