Theorem preq12 4680

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 4678	. 2 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
2		preq2 4679	. 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 4570

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 3432  df-un 3895  df-sn 4569  df-pr 4571

This theorem is referenced by:  preq12i  4683  preq12d  4686  ssprsseq  4769  preq12b  4794  prnebg  4800  preq12nebg  4807  opthprneg  4809  elpr2elpr  4813  relop  5797  opthreg  9528  hashle2pr  14428  wwlktovfo  14909  joinval  18330  meetval  18344  ipole  18489  sylow1  19567  frgpuplem  19736  uspgr2wlkeq  29734  wlkres  29757  wlkp1lem8  29767  usgr2pthlem  29851  2wlkdlem10  30023  1wlkdlem4  30230  3wlkdlem6  30255  3wlkdlem10  30259  pfxwlk  35327  oppr  47475  imarnf1pr  47727  elsprel  47932  sprsymrelf1lem  47948  sprsymrelf  47952  paireqne  47968  sbcpr  47978  isuspgrimlem  48368  grtrif1o  48415