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Theorem preq12i 4407
Description: Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Hypotheses
Ref Expression
preq1i.1 𝐴 = 𝐵
preq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
preq12i {𝐴, 𝐶} = {𝐵, 𝐷}

Proof of Theorem preq12i
StepHypRef Expression
1 preq1i.1 . 2 𝐴 = 𝐵
2 preq12i.2 . 2 𝐶 = 𝐷
3 preq12 4404 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷})
41, 2, 3mp2an 664 1 {𝐴, 𝐶} = {𝐵, 𝐷}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1630  {cpr 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-un 3726  df-sn 4315  df-pr 4317
This theorem is referenced by:  grpbasex  16201  grpplusgx  16202  indistpsx  21034  lgsdir2lem5  25274  wlk2v2elem2  27333  tgrpset  36547  zlmodzxzadd  42654  zlmodzxzequa  42803  zlmodzxzequap  42806
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