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Theorem nfop 3728
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 3540 . 2 𝐴, 𝐵⟩ = {𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
2 nfop.1 . . . . 5 𝑥𝐴
32nfel1 2293 . . . 4 𝑥 𝐴 ∈ V
4 nfop.2 . . . . 5 𝑥𝐵
54nfel1 2293 . . . 4 𝑥 𝐵 ∈ V
62nfsn 3590 . . . . . 6 𝑥{𝐴}
72, 4nfpr 3580 . . . . . 6 𝑥{𝐴, 𝐵}
86, 7nfpr 3580 . . . . 5 𝑥{{𝐴}, {𝐴, 𝐵}}
98nfcri 2276 . . . 4 𝑥 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}}
103, 5, 9nf3an 1546 . . 3 𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})
1110nfab 2287 . 2 𝑥{𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
121, 11nfcxfr 2279 1 𝑥𝐴, 𝐵
Colors of variables: wff set class
Syntax hints:  w3a 963  wcel 1481  {cab 2126  wnfc 2269  Vcvv 2689  {csn 3531  {cpr 3532  cop 3534
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538  df-op 3540
This theorem is referenced by:  nfopd  3729  moop2  4180  fliftfuns  5706  dfmpo  6127  qliftfuns  6520  xpf1o  6745  caucvgprprlemaddq  7539  nfseq  10258  txcnp  12477  cnmpt1t  12491  cnmpt2t  12499
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