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Theorem hbab1 2721
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2024.)
Assertion
Ref Expression
hbab1 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem hbab1
StepHypRef Expression
1 nfsab1 2720 . 2 𝑥 𝑦 ∈ {𝑥𝜑}
21nf5ri 2193 1 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535  wcel 2106  {cab 2712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713
This theorem is referenced by:  eqabbOLD  2880  bnj1317  34814  bnj1318  35018  bj-nfsab1  36799
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