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Theorem preqr1 4347
 Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.)
Hypotheses
Ref Expression
preqr1.a 𝐴 ∈ V
preqr1.b 𝐵 ∈ V
Assertion
Ref Expression
preqr1 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)

Proof of Theorem preqr1
StepHypRef Expression
1 preqr1.a . . 3 𝐴 ∈ V
2 id 22 . . . 4 (𝐴 ∈ V → 𝐴 ∈ V)
3 preqr1.b . . . . 5 𝐵 ∈ V
43a1i 11 . . . 4 (𝐴 ∈ V → 𝐵 ∈ V)
52, 4preq1b 4345 . . 3 (𝐴 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
61, 5ax-mp 5 . 2 ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵)
76biimpi 206 1 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1480   ∈ wcel 1987  Vcvv 3186  {cpr 4150 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-un 3560  df-sn 4149  df-pr 4151 This theorem is referenced by:  preqr2  4349  opthwiener  4936  cusgrfilem2  26239  usgr2wlkneq  26521  wopprc  37074
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