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Theorem preqr2 4321
 Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 15-Jul-1993.)
Hypotheses
Ref Expression
preqr1.a 𝐴 ∈ V
preqr1.b 𝐵 ∈ V
Assertion
Ref Expression
preqr2 ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)

Proof of Theorem preqr2
StepHypRef Expression
1 prcom 4211 . . 3 {𝐶, 𝐴} = {𝐴, 𝐶}
2 prcom 4211 . . 3 {𝐶, 𝐵} = {𝐵, 𝐶}
31, 2eqeq12i 2624 . 2 ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
4 preqr1.a . . 3 𝐴 ∈ V
5 preqr1.b . . 3 𝐵 ∈ V
64, 5preqr1 4319 . 2 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)
73, 6sylbi 206 1 ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {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:  preq12b  4322  opth  4871  opthreg  8398  usgra2edg  25911  usgraedgreu  25917  nbgraf1olem5  25974  altopthsn  31238  usgredgreu  40445  uspgredg2vtxeu  40447
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