Theorem preq12 4694

Description: Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Assertion

Ref	Expression
preq12	⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})

Proof of Theorem preq12

Step	Hyp	Ref	Expression
1		preq1 4692	. 2 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
2		preq2 4693	. 2 ⊢ (𝐵 = 𝐷 → {𝐶, 𝐵} = {𝐶, 𝐷})
3	1, 2	sylan9eq 2792	1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  {cpr 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908  df-sn 4583  df-pr 4585

This theorem is referenced by:  preq12i  4697  preq12d  4700  ssprsseq  4783  preq12b  4808  prnebg  4814  preq12nebg  4821  opthprneg  4823  elpr2elpr  4827  relop  5807  opthreg  9539  hashle2pr  14412  wwlktovfo  14893  joinval  18310  meetval  18324  ipole  18469  sylow1  19544  frgpuplem  19713  uspgr2wlkeq  29731  wlkres  29754  wlkp1lem8  29764  usgr2pthlem  29848  2wlkdlem10  30020  1wlkdlem4  30227  3wlkdlem6  30252  3wlkdlem10  30256  pfxwlk  35337  oppr  47387  imarnf1pr  47639  elsprel  47832  sprsymrelf1lem  47848  sprsymrelf  47852  paireqne  47868  sbcpr  47878  isuspgrimlem  48252  grtrif1o  48299