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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 43124. Version of nfra2 3349 with a disjoint variable condition not requiring ax-13 2370. (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 3191 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) | |
2 | nfa2 2170 | . 2 ⊢ Ⅎ𝑦∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) | |
3 | 1, 2 | nfxfr 1855 | 1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1539 Ⅎwnf 1785 ∈ wcel 2106 ∀wral 3064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-10 2137 ax-11 2154 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ex 1782 df-nf 1786 df-ral 3065 |
This theorem is referenced by: invdisj 5089 reusv3 5360 dedekind 11317 dedekindle 11318 mreexexd 17527 gsummatr01lem4 22005 ordtconnlem1 32496 bnj1379 33433 tratrb 42800 islptre 43832 sprsymrelfo 45661 |
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