Theorem preq12i 4699

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 4696	. 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 1570  {cpr 4586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 3904  df-sn 4585  df-pr 4587

This theorem is used by:  grpbasex  17378  grpplusgx  17379  indistpsx  23236  lgsdir2lem5  27566  neg1s  28293  wlk2v2elem2  30637  tgrpset  41619  nregmodelf1o  45839  stgr0  48877  stgr1  48878  gpgprismgr4cycllem10  49021  grlimedgnedg  49048  zlmodzxzadd  49289  zlmodzxzequa  49427  zlmodzxzequap  49430