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Theorem bnj1230 29920
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 3098 . . 3 𝑥{𝑥𝐴𝜑}
31, 2nfcxfr 2748 . 2 𝑥𝐵
43nfcrii 2743 1 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472   = wceq 1474  wcel 1976  {crab 2899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904
This theorem is referenced by:  bnj1312  30173
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