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Theorem bnj1441g 32808
Description: First-order logic and set theory. See bnj1441 32807 for a version with more disjoint variable conditions, but not requiring ax-13 2372. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1441g.1 (𝑥𝐴 → ∀𝑦 𝑥𝐴)
bnj1441g.2 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
bnj1441g (𝑧 ∈ {𝑥𝐴𝜑} → ∀𝑦 𝑧 ∈ {𝑥𝐴𝜑})
Distinct variable group:   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem bnj1441g
StepHypRef Expression
1 df-rab 3073 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 bnj1441g.1 . . . 4 (𝑥𝐴 → ∀𝑦 𝑥𝐴)
3 bnj1441g.2 . . . 4 (𝜑 → ∀𝑦𝜑)
42, 3hban 2297 . . 3 ((𝑥𝐴𝜑) → ∀𝑦(𝑥𝐴𝜑))
54hbabg 2727 . 2 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∀𝑦 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
61, 5hbxfreq 2869 1 (𝑧 ∈ {𝑥𝐴𝜑} → ∀𝑦 𝑧 ∈ {𝑥𝐴𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1537  wcel 2106  {cab 2715  {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-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073
This theorem is referenced by: (None)
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