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Theorem preqr1 4764
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 4762 . . 3 (𝐴 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
61, 5ax-mp 5 . 2 ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵)
76biimpi 219 1 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2110  Vcvv 3413  {cpr 4548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3415  df-un 3876  df-sn 4547  df-pr 4549
This theorem is referenced by:  preqr2  4765  opthwiener  5402  opthhausdorff0  5406  cusgrfilem2  27549  usgr2wlkneq  27848  wopprc  40563
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