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Theorem preqr2 4783
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 4671 . . 3 {𝐶, 𝐴} = {𝐴, 𝐶}
2 prcom 4671 . . 3 {𝐶, 𝐵} = {𝐵, 𝐶}
31, 2eqeq12i 2839 . 2 ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
4 preqr1.a . . 3 𝐴 ∈ V
5 preqr1.b . . 3 𝐵 ∈ V
64, 5preqr1 4782 . 2 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)
73, 6sylbi 219 1 ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2113  Vcvv 3497  {cpr 4572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-un 3944  df-sn 4571  df-pr 4573
This theorem is referenced by:  preq12b  4784  opth  5371  opthreg  9084  usgredgreu  27003  uspgredg2vtxeu  27005  altopthsn  33426
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