New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  nfopab2 GIF version

Theorem nfopab2 4629
 Description: The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
nfopab2 y{x, y φ}

Proof of Theorem nfopab2
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-opab 4623 . 2 {x, y φ} = {z xy(z = x, y φ)}
2 nfe1 1732 . . . 4 yy(z = x, y φ)
32nfex 1843 . . 3 yxy(z = x, y φ)
43nfab 2493 . 2 y{z xy(z = x, y φ)}
51, 4nfcxfr 2486 1 y{x, y φ}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642  {cab 2339  Ⅎwnfc 2476  ⟨cop 4561  {copab 4622 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-opab 4623 This theorem is referenced by:  opelopabsb  4697  ssopab2b  4713  dmopab  4915  rnopab  4967  funopab  5139
 Copyright terms: Public domain W3C validator