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Theorem bnj1441 32120
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Add disjoint variable condition to avoid ax-13 2391. See bnj1441g 32121 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 3135 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 bnj1441.1 . . . 4 (𝑥𝐴 → ∀𝑦 𝑥𝐴)
3 bnj1441.2 . . . 4 (𝜑 → ∀𝑦𝜑)
42, 3hban 2309 . . 3 ((𝑥𝐴𝜑) → ∀𝑦(𝑥𝐴𝜑))
54hbab 2810 . 2 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∀𝑦 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
61, 5hbxfreq 2941 1 (𝑧 ∈ {𝑥𝐴𝜑} → ∀𝑦 𝑧 ∈ {𝑥𝐴𝜑})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   ∈ wcel 2115  {cab 2799  {crab 3130 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-rab 3135 This theorem is referenced by: (None)
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