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Theorem nfop 4393
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 4374 . 2 𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
2 nfop.1 . . . . 5 𝑥𝐴
32nfel1 2775 . . . 4 𝑥 𝐴 ∈ V
4 nfop.2 . . . . 5 𝑥𝐵
54nfel1 2775 . . . 4 𝑥 𝐵 ∈ V
63, 5nfan 1825 . . 3 𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V)
72nfsn 4220 . . . 4 𝑥{𝐴}
82, 4nfpr 4210 . . . 4 𝑥{𝐴, 𝐵}
97, 8nfpr 4210 . . 3 𝑥{{𝐴}, {𝐴, 𝐵}}
10 nfcv 2761 . . 3 𝑥
116, 9, 10nfif 4093 . 2 𝑥if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
121, 11nfcxfr 2759 1 𝑥𝐴, 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 384  wcel 1987  wnfc 2748  Vcvv 3190  c0 3897  ifcif 4064  {csn 4155  {cpr 4157  cop 4161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162
This theorem is referenced by:  nfopd  4394  moop2  4936  iunopeqop  4951  fliftfuns  6529  dfmpt2  7227  qliftfuns  7794  xpf1o  8082  nfseq  12767  txcnp  21363  cnmpt1t  21408  cnmpt2t  21416  flfcnp2  21751  bnj958  30771  bnj1000  30772  bnj1446  30874  bnj1447  30875  bnj1448  30876  bnj1466  30882  bnj1467  30883  bnj1519  30894  bnj1520  30895  bnj1529  30899  poimirlem26  33106  nfopdALT  33777  nfaov  40593
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