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Theorem nfop 4889
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 4870 . 2 𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
2 nfop.1 . . . . 5 𝑥𝐴
32nfel1 2922 . . . 4 𝑥 𝐴 ∈ V
4 nfop.2 . . . . 5 𝑥𝐵
54nfel1 2922 . . . 4 𝑥 𝐵 ∈ V
63, 5nfan 1899 . . 3 𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V)
72nfsn 4707 . . . 4 𝑥{𝐴}
82, 4nfpr 4692 . . . 4 𝑥{𝐴, 𝐵}
97, 8nfpr 4692 . . 3 𝑥{{𝐴}, {𝐴, 𝐵}}
10 nfcv 2905 . . 3 𝑥
116, 9, 10nfif 4556 . 2 𝑥if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
121, 11nfcxfr 2903 1 𝑥𝐴, 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 395  wcel 2108  wnfc 2890  Vcvv 3480  c0 4333  ifcif 4525  {csn 4626  {cpr 4628  cop 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633
This theorem is referenced by:  nfopd  4890  moop2  5507  iunopeqop  5526  fliftfuns  7334  dfmpo  8127  qliftfuns  8844  xpf1o  9179  nfseq  14052  txcnp  23628  cnmpt1t  23673  cnmpt2t  23681  flfcnp2  24015  nosupbnd2  27761  noinfbnd2  27776  nfseqs  28293  bnj958  34954  bnj1000  34955  bnj1446  35059  bnj1447  35060  bnj1448  35061  bnj1466  35067  bnj1467  35068  bnj1519  35079  bnj1520  35080  bnj1529  35084  poimirlem26  37653  nfopdALT  38972  nfaov  47191
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