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Theorem nfra2w 3282
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.)
Assertion
Ref Expression
nfra2w 𝑦𝑥𝐴𝑦𝐵 𝜑
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem nfra2w
StepHypRef Expression
1 r2al 3191 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
2 nfa2 2170 . 2 𝑦𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑)
31, 2nfxfr 1855 1 𝑦𝑥𝐴𝑦𝐵 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539  wnf 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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