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Theorem nfopab1 3965
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 3958 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfe1 1455 . . 3 𝑥𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
32nfab 2261 . 2 𝑥{𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
41, 3nfcxfr 2253 1 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1314  wex 1451  {cab 2101  wnfc 2243  cop 3498  {copab 3956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-opab 3958
This theorem is referenced by:  nfmpt1  3989  opelopabsb  4150  ssopab2b  4166  dmopab  4718  rnopab  4754  funopab  5126  0neqopab  5782
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