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Theorem preqr2 4525
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 4411 . . 3 {𝐶, 𝐴} = {𝐴, 𝐶}
2 prcom 4411 . . 3 {𝐶, 𝐵} = {𝐵, 𝐶}
31, 2eqeq12i 2774 . 2 ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
4 preqr1.a . . 3 𝐴 ∈ V
5 preqr1.b . . 3 𝐵 ∈ V
64, 5preqr1 4524 . 2 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)
73, 6sylbi 207 1 ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  {cpr 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-un 3720  df-sn 4322  df-pr 4324
This theorem is referenced by:  preq12b  4526  opth  5093  opthreg  8686  opthregOLD  8688  usgredgreu  26309  uspgredg2vtxeu  26311  altopthsn  32374
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