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Theorem bnj1230 32782
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1230.1 𝐵 = {𝑥𝐴𝜑}
Assertion
Ref Expression
bnj1230 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem bnj1230
StepHypRef Expression
1 bnj1230.1 . . 3 𝐵 = {𝑥𝐴𝜑}
2 nfrab1 3317 . . 3 𝑥{𝑥𝐴𝜑}
31, 2nfcxfr 2905 . 2 𝑥𝐵
43nfcrii 2899 1 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1537   = wceq 1539  wcel 2106  {crab 3068
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073
This theorem is referenced by:  bnj1312  33038
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