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Theorem hbab 2080
 Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)
Hypothesis
Ref Expression
hbab.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbab (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})
Distinct variable group:   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem hbab
StepHypRef Expression
1 df-clab 2076 . 2 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
2 hbab.1 . . 3 (𝜑 → ∀𝑥𝜑)
32hbsb 1872 . 2 ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)
41, 3hbxfrbi 1407 1 (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1288   ∈ wcel 1439  [wsb 1693  {cab 2075 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474 This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076 This theorem is referenced by:  nfsab  2081
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