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Theorem nfsab 2643
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
nfsab.1 𝑥𝜑
Assertion
Ref Expression
nfsab 𝑥 𝑧 ∈ {𝑦𝜑}
Distinct variable group:   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsab
StepHypRef Expression
1 nfsab.1 . . . 4 𝑥𝜑
21nf5ri 2103 . . 3 (𝜑 → ∀𝑥𝜑)
32hbab 2642 . 2 (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})
43nf5i 2064 1 𝑥 𝑧 ∈ {𝑦𝜑}
Colors of variables: wff setvar class
Syntax hints:  wnf 1748  wcel 2030  {cab 2637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638
This theorem is referenced by:  nfab  2798  upbdrech  39833  ssfiunibd  39837
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