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Theorem nfopab1 4628
 Description: The first 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
nfopab1 x{x, y φ}

Proof of Theorem nfopab1
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 . . 3 xxy(z = x, y φ)
32nfab 2493 . 2 x{z xy(z = x, y φ)}
41, 3nfcxfr 2486 1 x{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  nfmpt1  5672
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