MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfsab1 Structured version   Visualization version   GIF version

Theorem nfsab1 2761
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
Assertion
Ref Expression
nfsab1 𝑥 𝑦 ∈ {𝑥𝜑}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem nfsab1
StepHypRef Expression
1 hbab1 2760 . 2 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
21nf5i 2179 1 𝑥 𝑦 ∈ {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wnf 1856  wcel 2145  {cab 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758
This theorem is referenced by:  clelab  2897  nfab1  2915  ralab2  3523  rexab2  3525  eluniab  4585  elintab  4622  opabex3d  7292  opabex3  7293  setindtrs  38115  rababg  38402
  Copyright terms: Public domain W3C validator