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Theorem grothprimlem 9534
 Description: Lemma for grothprim 9535. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.)
Assertion
Ref Expression
grothprimlem ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
Distinct variable group:   𝑤,𝑣,𝑢,,𝑔

Proof of Theorem grothprimlem
StepHypRef Expression
1 dfpr2 4143 . . 3 {𝑢, 𝑣} = { ∣ ( = 𝑢 = 𝑣)}
21eleq1i 2679 . 2 ({𝑢, 𝑣} ∈ 𝑤 ↔ { ∣ ( = 𝑢 = 𝑣)} ∈ 𝑤)
3 clabel 2736 . 2 ({ ∣ ( = 𝑢 = 𝑣)} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
42, 3bitri 263 1 ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔𝑤 ∧ ∀(𝑔 ↔ ( = 𝑢 = 𝑣))))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∨ wo 382   ∧ wa 383  ∀wal 1473  ∃wex 1695   ∈ wcel 1977  {cab 2596  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128 This theorem is referenced by:  grothprim  9535
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