Theorem preq12i 4697

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 4694	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷})
4	1, 2, 3	mp2an 702	1 ⊢ {𝐴, 𝐶} = {𝐵, 𝐷}

Syntax hints:   = wceq 1560  {cpr 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-sn 4583  df-pr 4585

This theorem is referenced by:  grpbasex  17321  grpplusgx  17322  indistpsx  23070  lgsdir2lem5  27393  neg1s  28120  wlk2v2elem2  30358  tgrpset  41369  nregmodelf1o  45591  stgr0  48582  stgr1  48583  gpgprismgr4cycllem10  48726  grlimedgnedg  48753  zlmodzxzadd  48980  zlmodzxzequa  49118  zlmodzxzequap  49121