Theorem preq12i 4702

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

Step	Hyp	Ref	Expression
1		preq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		preq12i.2	. 2 ⊢ 𝐶 = 𝐷
3		preq12 4699	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷})
4	1, 2, 3	mp2an 692	1 ⊢ {𝐴, 𝐶} = {𝐵, 𝐷}

Syntax hints:   = wceq 1540  {cpr 4591

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-un 3919  df-sn 4590  df-pr 4592

This theorem is referenced by:  grpbasex  17255  grpplusgx  17256  indistpsx  22897  lgsdir2lem5  27240  negs1s  27933  wlk2v2elem2  30085  tgrpset  40739  nregmodelf1o  45005  stgr0  47959  stgr1  47960  gpgprismgr4cycllem10  48094  zlmodzxzadd  48346  zlmodzxzequa  48485  zlmodzxzequap  48488