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Theorem nfpr 3668
Description: Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
Hypotheses
Ref Expression
nfpr.1 𝑥𝐴
nfpr.2 𝑥𝐵
Assertion
Ref Expression
nfpr 𝑥{𝐴, 𝐵}

Proof of Theorem nfpr
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfpr2 3637 . 2 {𝐴, 𝐵} = {𝑦 ∣ (𝑦 = 𝐴𝑦 = 𝐵)}
2 nfpr.1 . . . . 5 𝑥𝐴
32nfeq2 2348 . . . 4 𝑥 𝑦 = 𝐴
4 nfpr.2 . . . . 5 𝑥𝐵
54nfeq2 2348 . . . 4 𝑥 𝑦 = 𝐵
63, 5nfor 1585 . . 3 𝑥(𝑦 = 𝐴𝑦 = 𝐵)
76nfab 2341 . 2 𝑥{𝑦 ∣ (𝑦 = 𝐴𝑦 = 𝐵)}
81, 7nfcxfr 2333 1 𝑥{𝐴, 𝐵}
Colors of variables: wff set class
Syntax hints:  wo 709   = wceq 1364  {cab 2179  wnfc 2323  {cpr 3619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625
This theorem is referenced by:  nfsn  3678  nfop  3820
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