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Theorem nfpr 4692
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 4646 . 2 {𝐴, 𝐵} = {𝑦 ∣ (𝑦 = 𝐴𝑦 = 𝐵)}
2 nfpr.1 . . . . 5 𝑥𝐴
32nfeq2 2923 . . . 4 𝑥 𝑦 = 𝐴
4 nfpr.2 . . . . 5 𝑥𝐵
54nfeq2 2923 . . . 4 𝑥 𝑦 = 𝐵
63, 5nfor 1904 . . 3 𝑥(𝑦 = 𝐴𝑦 = 𝐵)
76nfab 2911 . 2 𝑥{𝑦 ∣ (𝑦 = 𝐴𝑦 = 𝐵)}
81, 7nfcxfr 2903 1 𝑥{𝐴, 𝐵}
Colors of variables: wff setvar class
Syntax hints:  wo 848   = wceq 1540  {cab 2714  wnfc 2890  {cpr 4628
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-tru 1543  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-un 3956  df-sn 4627  df-pr 4629
This theorem is referenced by:  nfsn  4707  nfop  4889  nfaltop  35981
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