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Theorem preqr2 3691
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 3594 . . 3 {𝐶, 𝐴} = {𝐴, 𝐶}
2 prcom 3594 . . 3 {𝐶, 𝐵} = {𝐵, 𝐶}
31, 2eqeq12i 2151 . 2 ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
4 preqr2.1 . . 3 𝐴 ∈ V
5 preqr2.2 . . 3 𝐵 ∈ V
64, 5preqr1 3690 . 2 ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)
73, 6sylbi 120 1 ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  Vcvv 2681  {cpr 3523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529
This theorem is referenced by:  preq12b  3692  opth  4154  opthreg  4466
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