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Theorem nfopab2 5103
 Description: The second 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
nfopab2 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem nfopab2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-opab 5096 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfe1 2152 . . . 4 𝑦𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
32nfex 2333 . . 3 𝑦𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
43nfab 2926 . 2 𝑦{𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
51, 4nfcxfr 2918 1 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 400   = wceq 1539  ∃wex 1782  {cab 2736  Ⅎwnfc 2900  ⟨cop 4529  {copab 5095 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-opab 5096 This theorem is referenced by:  rexopabb  5386  ssopab2bw  5405  ssopab2b  5407  0nelopab  5423  dmopab  5756  rnopab  5796  funopab  6371  fvopab5  6792  zfrep6  7661  opabdm  30474  opabrn  30475  fpwrelmap  30593  bj-opabco  34884  vvdifopab  35962  aomclem8  40379  areaquad  40540  sprsymrelf  44381
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