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Theorem nfop 3593
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 3412 . 2 𝐴, 𝐵⟩ = {𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
2 nfop.1 . . . . 5 𝑥𝐴
32nfel1 2204 . . . 4 𝑥 𝐴 ∈ V
4 nfop.2 . . . . 5 𝑥𝐵
54nfel1 2204 . . . 4 𝑥 𝐵 ∈ V
62nfsn 3458 . . . . . 6 𝑥{𝐴}
72, 4nfpr 3448 . . . . . 6 𝑥{𝐴, 𝐵}
86, 7nfpr 3448 . . . . 5 𝑥{{𝐴}, {𝐴, 𝐵}}
98nfcri 2188 . . . 4 𝑥 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}}
103, 5, 9nf3an 1474 . . 3 𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})
1110nfab 2198 . 2 𝑥{𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
121, 11nfcxfr 2191 1 𝑥𝐴, 𝐵
Colors of variables: wff set class
Syntax hints:  w3a 896  wcel 1409  {cab 2042  wnfc 2181  Vcvv 2574  {csn 3403  {cpr 3404  cop 3406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412
This theorem is referenced by:  nfopd  3594  moop2  4016  fliftfuns  5466  dfmpt2  5872  qliftfuns  6221  caucvgprprlemaddq  6864  nfiseq  9382
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