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Theorem nfpr 4656
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 4610 . 2 {𝐴, 𝐵} = {𝑦 ∣ (𝑦 = 𝐴𝑦 = 𝐵)}
2 nfpr.1 . . . . 5 𝑥𝐴
32nfeq2 2925 . . . 4 𝑥 𝑦 = 𝐴
4 nfpr.2 . . . . 5 𝑥𝐵
54nfeq2 2925 . . . 4 𝑥 𝑦 = 𝐵
63, 5nfor 1908 . . 3 𝑥(𝑦 = 𝐴𝑦 = 𝐵)
76nfab 2914 . 2 𝑥{𝑦 ∣ (𝑦 = 𝐴𝑦 = 𝐵)}
81, 7nfcxfr 2906 1 𝑥{𝐴, 𝐵}
Colors of variables: wff setvar class
Syntax hints:  wo 846   = wceq 1542  {cab 2714  wnfc 2888  {cpr 4593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-v 3450  df-un 3920  df-sn 4592  df-pr 4594
This theorem is referenced by:  nfsn  4673  nfop  4851  nfaltop  34594
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