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Theorem bnj1230 31973
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 3382 . . 3 𝑥{𝑥𝐴𝜑}
31, 2nfcxfr 2972 . 2 𝑥𝐵
43nfcrii 2967 1 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526   = wceq 1528  wcel 2105  {crab 3139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144
This theorem is referenced by:  bnj1312  32227
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