Theorem preq12i 4695

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 4692	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷})
4	1, 2, 3	mp2an 705	1 ⊢ {𝐴, 𝐶} = {𝐵, 𝐷}

This proof depends on syntax axioms:   = wceq 1569  {cpr 4586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 4150  df-sn 4585  df-pr 4587

This theorem is used by:  grpbasex  17472  grpplusgx  17473  indistpsx  23337  lgsdir2lem5  27665  neg1s  28422  wlk2v2elem2  30766  tgrpset  41939  nregmodelf1o  46141  stgr0  49194  stgr1  49195  gpgprismgr4cycllem10  49338  grlimedgnedg  49365  zlmodzxzadd  49606  zlmodzxzequa  49744  zlmodzxzequap  49747