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Theorem nfop 3668
Description: Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
Hypotheses
Ref Expression
nfop.1 𝑥𝐴
nfop.2 𝑥𝐵
Assertion
Ref Expression
nfop 𝑥𝐴, 𝐵

Proof of Theorem nfop
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-op 3483 . 2 𝐴, 𝐵⟩ = {𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
2 nfop.1 . . . . 5 𝑥𝐴
32nfel1 2251 . . . 4 𝑥 𝐴 ∈ V
4 nfop.2 . . . . 5 𝑥𝐵
54nfel1 2251 . . . 4 𝑥 𝐵 ∈ V
62nfsn 3530 . . . . . 6 𝑥{𝐴}
72, 4nfpr 3520 . . . . . 6 𝑥{𝐴, 𝐵}
86, 7nfpr 3520 . . . . 5 𝑥{{𝐴}, {𝐴, 𝐵}}
98nfcri 2234 . . . 4 𝑥 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}}
103, 5, 9nf3an 1513 . . 3 𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})
1110nfab 2245 . 2 𝑥{𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
121, 11nfcxfr 2237 1 𝑥𝐴, 𝐵
Colors of variables: wff set class
Syntax hints:  w3a 930  wcel 1448  {cab 2086  wnfc 2227  Vcvv 2641  {csn 3474  {cpr 3475  cop 3477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483
This theorem is referenced by:  nfopd  3669  moop2  4111  fliftfuns  5631  dfmpo  6050  qliftfuns  6443  xpf1o  6667  caucvgprprlemaddq  7417  nfseq  10069  txcnp  12221  cnmpt1t  12235  cnmpt2t  12243
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