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Theorem nfop 4893
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
StepHypRef Expression
1 dfopif 4874 . 2 𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
2 nfop.1 . . . . 5 𝑥𝐴
32nfel1 2919 . . . 4 𝑥 𝐴 ∈ V
4 nfop.2 . . . . 5 𝑥𝐵
54nfel1 2919 . . . 4 𝑥 𝐵 ∈ V
63, 5nfan 1896 . . 3 𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V)
72nfsn 4711 . . . 4 𝑥{𝐴}
82, 4nfpr 4696 . . . 4 𝑥{𝐴, 𝐵}
97, 8nfpr 4696 . . 3 𝑥{{𝐴}, {𝐴, 𝐵}}
10 nfcv 2902 . . 3 𝑥
116, 9, 10nfif 4560 . 2 𝑥if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
121, 11nfcxfr 2900 1 𝑥𝐴, 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 395  wcel 2105  wnfc 2887  Vcvv 3477  c0 4338  ifcif 4530  {csn 4630  {cpr 4632  cop 4636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637
This theorem is referenced by:  nfopd  4894  moop2  5511  iunopeqop  5530  fliftfuns  7333  dfmpo  8125  qliftfuns  8842  xpf1o  9177  nfseq  14048  txcnp  23643  cnmpt1t  23688  cnmpt2t  23696  flfcnp2  24030  nosupbnd2  27775  noinfbnd2  27790  nfseqs  28307  bnj958  34932  bnj1000  34933  bnj1446  35037  bnj1447  35038  bnj1448  35039  bnj1466  35045  bnj1467  35046  bnj1519  35057  bnj1520  35058  bnj1529  35062  poimirlem26  37632  nfopdALT  38952  nfaov  47128
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