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Theorem hbabg 2748
 Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2380. See hbab 2747 for a version with more disjoint variable conditions, but not requiring ax-13 2380. (Contributed by NM, 1-Mar-1995.) (New usage is discouraged.)
Hypothesis
Ref Expression
hbabg.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbabg (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})
Distinct variable group:   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem hbabg
StepHypRef Expression
1 df-clab 2737 . 2 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
2 hbabg.1 . . 3 (𝜑 → ∀𝑥𝜑)
32hbsb 2545 . 2 ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)
41, 3hbxfrbi 1827 1 (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1537  [wsb 2070   ∈ wcel 2112  {cab 2736 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2380 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737 This theorem is referenced by:  nfsabg  2750  bnj1441g  32334
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