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Theorem hbab 2747
 Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) Add disjoint variable condition to avoid ax-13 2380. See hbabg 2748 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Hypothesis
Ref Expression
hbab.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbab (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})
Distinct variable groups:   𝑥,𝑦   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem hbab
StepHypRef Expression
1 df-clab 2737 . 2 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
2 hbab.1 . . 3 (𝜑 → ∀𝑥𝜑)
32hbsbw 2174 . 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-11 2159 This theorem depends on definitions:  df-bi 210  df-ex 1783  df-sb 2071  df-clab 2737 This theorem is referenced by:  nfsab  2749  bnj1441  32341  bnj1309  32523
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