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Theorem hbab1 2767
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.)
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 df-clab 2760 . 2 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
2 hbs1 2203 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
31, 2hbxfrbi 1787 1 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1505  [wsb 2015  wcel 2050  {cab 2759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-12 2106
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760
This theorem is referenced by:  nfsab1OLD  2769  abeq2  2898  abbi  2908  abeq2fOLD  2966  bnj1317  31738  bnj1318  31939
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