![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 43669. Version of nfra2 3373 with a disjoint variable condition not requiring ax-13 2372. (Contributed by Alan Sare, 31-Dec-2011.) Reduce axiom usage. (Revised by Gino Giotto, 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 3195 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) | |
2 | nfa2 2171 | . 2 ⊢ Ⅎ𝑦∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) | |
3 | 1, 2 | nfxfr 1856 | 1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1540 Ⅎwnf 1786 ∈ wcel 2107 ∀wral 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-10 2138 ax-11 2155 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ex 1783 df-nf 1787 df-ral 3063 |
This theorem is referenced by: invdisj 5133 reusv3 5404 dedekind 11377 dedekindle 11378 mreexexd 17592 gsummatr01lem4 22160 ordtconnlem1 32935 bnj1379 33872 tratrb 43345 islptre 44383 sprsymrelfo 46213 |
Copyright terms: Public domain | W3C validator |