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Theorem nfra2w 3297
Description: Similar to Lemma 24 of [Monk2] p. 114, except that quantification is restricted. Once derived from hbra2VD 44858. Version of nfra2 3374 with a disjoint variable condition not requiring ax-13 2375. (Contributed by Alan Sare, 31-Dec-2011.) Reduce axiom usage. (Revised by GG, 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 3193 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
2 nfa2 2174 . 2 𝑦𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑)
31, 2nfxfr 1850 1 𝑦𝑥𝐴𝑦𝐵 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535  wnf 1780  wcel 2106  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-10 2139  ax-11 2155
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-ral 3060
This theorem is referenced by:  invdisj  5134  reusv3  5411  dedekind  11422  dedekindle  11423  mreexexd  17693  gsummatr01lem4  22680  ordtconnlem1  33885  bnj1379  34823  tratrb  44534  islptre  45575  sprsymrelfo  47422
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