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Theorem hbab1 2784
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 2777 . 2 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
2 hbs1 2271 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
31, 2hbxfrbi 1826 1 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  [wsb 2069  wcel 2111  {cab 2776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777
This theorem is referenced by:  nfsab1OLD  2786  abeq2  2922  abbiOLD  2930  bnj1317  32203  bnj1318  32407  bj-nfsab1  34250
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