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Theorem bnj1230 32182
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 3340 . . 3 𝑥{𝑥𝐴𝜑}
31, 2nfcxfr 2956 . 2 𝑥𝐵
43nfcrii 2951 1 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536   = wceq 1538  wcel 2112  {crab 3113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118
This theorem is referenced by:  bnj1312  32438
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