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Theorem preqr2 3787
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
preqr2.1 𝐴 ∈ V
preqr2.2 𝐵 ∈ V
Assertion
Ref Expression
preqr2 ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)

Proof of Theorem preqr2
StepHypRef Expression
1 prcom 3686 . . 3 {𝐶, 𝐴} = {𝐴, 𝐶}
2 prcom 3686 . . 3 {𝐶, 𝐵} = {𝐵, 𝐶}
31, 2eqeq12i 2203 . 2 ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
4 preqr2.1 . . 3 𝐴 ∈ V
5 preqr2.2 . . 3 𝐵 ∈ V
64, 5preqr1 3786 . 2 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)
73, 6sylbi 121 1 ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  Vcvv 2752  {cpr 3611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3616  df-pr 3617
This theorem is referenced by:  preq12b  3788  opth  4258  opthreg  4576
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