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Theorem bnj1230 30999
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 3152 . . 3 𝑥{𝑥𝐴𝜑}
31, 2nfcxfr 2791 . 2 𝑥𝐵
43nfcrii 2786 1 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521   = wceq 1523  wcel 2030  {crab 2945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950
This theorem is referenced by:  bnj1312  31252
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