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Theorem preqr1 4509
 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 4507 . . 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 1630   ∈ wcel 2144  Vcvv 3349  {cpr 4316 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-un 3726  df-sn 4315  df-pr 4317 This theorem is referenced by:  preqr2  4510  opthwiener  5107  opthhausdorff0  5111  cusgrfilem2  26586  usgr2wlkneq  26886  wopprc  38116
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