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Theorem nfop 3691
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 3506 . 2 𝐴, 𝐵⟩ = {𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
2 nfop.1 . . . . 5 𝑥𝐴
32nfel1 2269 . . . 4 𝑥 𝐴 ∈ V
4 nfop.2 . . . . 5 𝑥𝐵
54nfel1 2269 . . . 4 𝑥 𝐵 ∈ V
62nfsn 3553 . . . . . 6 𝑥{𝐴}
72, 4nfpr 3543 . . . . . 6 𝑥{𝐴, 𝐵}
86, 7nfpr 3543 . . . . 5 𝑥{{𝐴}, {𝐴, 𝐵}}
98nfcri 2252 . . . 4 𝑥 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}}
103, 5, 9nf3an 1530 . . 3 𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})
1110nfab 2263 . 2 𝑥{𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
121, 11nfcxfr 2255 1 𝑥𝐴, 𝐵
Colors of variables: wff set class
Syntax hints:  w3a 947  wcel 1465  {cab 2103  wnfc 2245  Vcvv 2660  {csn 3497  {cpr 3498  cop 3500
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506
This theorem is referenced by:  nfopd  3692  moop2  4143  fliftfuns  5667  dfmpo  6088  qliftfuns  6481  xpf1o  6706  caucvgprprlemaddq  7484  nfseq  10196  txcnp  12367  cnmpt1t  12381  cnmpt2t  12389
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