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Theorem nfra2w 3281
Description: Similar to Lemma 24 of [Monk2] p. 114, except that quantification is restricted. Once derived from hbra2VD 43230. Version of nfra2 3348 with a disjoint variable condition not requiring ax-13 2371. (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 3188 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
2 nfa2 2171 . 2 𝑦𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑)
31, 2nfxfr 1856 1 𝑦𝑥𝐴𝑦𝐵 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540  wnf 1786  wcel 2107  wral 3061
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 3062
This theorem is referenced by:  invdisj  5090  reusv3  5361  dedekind  11323  dedekindle  11324  mreexexd  17533  gsummatr01lem4  22023  ordtconnlem1  32562  bnj1379  33499  tratrb  42906  islptre  43946  sprsymrelfo  45775
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