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Theorem nfopab1 5218
Description: The first abstraction variable in an ordered-pair class abstraction 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 5211 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfe1 2148 . . 3 𝑥𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
32nfab 2909 . 2 𝑥{𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
41, 3nfcxfr 2901 1 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wex 1776  {cab 2712  wnfc 2888  cop 4637  {copab 5210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-opab 5211
This theorem is referenced by:  nfmpt1  5256  rexopabb  5538  ssopab2bw  5557  ssopab2b  5559  dmopab  5929  rnopab  5968  funopab  6603  fvopab5  7049  zfrep6  7978  opabdm  32631  opabrn  32632  fpwrelmap  32751  fineqvrep  35088  bj-opabco  37171  vvdifopab  38242  aomclem8  43050  sprsymrelf  47420
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