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Theorem nfpr 4535
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 4491 . 2 {𝐴, 𝐵} = {𝑦 ∣ (𝑦 = 𝐴𝑦 = 𝐵)}
2 nfpr.1 . . . . 5 𝑥𝐴
32nfeq2 2964 . . . 4 𝑥 𝑦 = 𝐴
4 nfpr.2 . . . . 5 𝑥𝐵
54nfeq2 2964 . . . 4 𝑥 𝑦 = 𝐵
63, 5nfor 1886 . . 3 𝑥(𝑦 = 𝐴𝑦 = 𝐵)
76nfab 2955 . 2 𝑥{𝑦 ∣ (𝑦 = 𝐴𝑦 = 𝐵)}
81, 7nfcxfr 2947 1 𝑥{𝐴, 𝐵}
Colors of variables: wff setvar class
Syntax hints:  wo 842   = wceq 1522  {cab 2775  wnfc 2933  {cpr 4474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-un 3864  df-sn 4473  df-pr 4475
This theorem is referenced by:  nfsn  4550  nfop  4726  nfaltop  33050
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