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Theorem hbab1 2029
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-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 2027 . 2 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
2 hbs1 1814 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
31, 2hbxfrbi 1361 1 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1241  wcel 1393  [wsb 1645  {cab 2026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-sb 1646  df-clab 2027
This theorem is referenced by:  nfsab1  2030  abeq2  2146
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