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Theorem bnj1441 34379
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Add disjoint variable condition to avoid ax-13 2365. See bnj1441g 34380 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) (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 groups:   𝑥,𝑦   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem bnj1441
StepHypRef Expression
1 df-rab 3427 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 bnj1441.1 . . . 4 (𝑥𝐴 → ∀𝑦 𝑥𝐴)
3 bnj1441.2 . . . 4 (𝜑 → ∀𝑦𝜑)
42, 3hban 2288 . . 3 ((𝑥𝐴𝜑) → ∀𝑦(𝑥𝐴𝜑))
54hbab 2714 . 2 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∀𝑦 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
61, 5hbxfreq 2858 1 (𝑧 ∈ {𝑥𝐴𝜑} → ∀𝑦 𝑧 ∈ {𝑥𝐴𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1531  wcel 2098  {cab 2703  {crab 3426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427
This theorem is referenced by: (None)
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