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Theorem bnj1441 31359
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 3064 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 bnj1441.1 . . . 4 (𝑥𝐴 → ∀𝑦 𝑥𝐴)
3 bnj1441.2 . . . 4 (𝜑 → ∀𝑦𝜑)
42, 3hban 2305 . . 3 ((𝑥𝐴𝜑) → ∀𝑦(𝑥𝐴𝜑))
54hbab 2756 . 2 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∀𝑦 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
61, 5hbxfreq 2873 1 (𝑧 ∈ {𝑥𝐴𝜑} → ∀𝑦 𝑧 ∈ {𝑥𝐴𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1650  wcel 2155  {cab 2751  {crab 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-rab 3064
This theorem is referenced by: (None)
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