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  5799  opthreg  9530  hashle2pr  14430  wwlktovfo  14911  joinval  18332  meetval  18346  ipole  18491  sylow1  19569  frgpuplem  19738  uspgr2wlkeq  29729  wlkres  29752  wlkp1lem8  29762  usgr2pthlem  29846  2wlkdlem10  30018  1wlkdlem4  30225  3wlkdlem6  30250  3wlkdlem10  30254  pfxwlk  35322  oppr  47490  imarnf1pr  47742  elsprel  47947  sprsymrelf1lem  47963  sprsymrelf  47967  paireqne  47983  sbcpr  47993  isuspgrimlem  48383  grtrif1o  48430