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Theorem bnj1441 30616
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1441.1 (𝑥𝐴 → ∀𝑦 𝑥𝐴)
bnj1441.2 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
bnj1441 (𝑧 ∈ {𝑥𝐴𝜑} → ∀𝑦 𝑧 ∈ {𝑥𝐴𝜑})
Distinct variable group:   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem bnj1441
StepHypRef Expression
1 df-rab 2916 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 bnj1441.1 . . . 4 (𝑥𝐴 → ∀𝑦 𝑥𝐴)
3 bnj1441.2 . . . 4 (𝜑 → ∀𝑦𝜑)
42, 3hban 2124 . . 3 ((𝑥𝐴𝜑) → ∀𝑦(𝑥𝐴𝜑))
54hbab 2612 . 2 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∀𝑦 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
61, 5hbxfreq 2727 1 (𝑧 ∈ {𝑥𝐴𝜑} → ∀𝑦 𝑧 ∈ {𝑥𝐴𝜑})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1478   ∈ wcel 1987  {cab 2607  {crab 2911 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-rab 2916 This theorem is referenced by: (None)
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