Theorem preq12 4665

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 4663	. 2 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
2		preq2 4664	. 2 ⊢ (𝐵 = 𝐷 → {𝐶, 𝐵} = {𝐶, 𝐷})
3	1, 2	sylan9eq 2876	1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533  {cpr 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-un 3941  df-sn 4562  df-pr 4564

This theorem is referenced by:  preq12i  4668  preq12d  4671  ssprsseq  4752  preq12b  4775  prnebg  4780  preq12nebg  4787  opthprneg  4789  snex  5324  relop  5716  opthreg  9075  hashle2pr  13829  wwlktovfo  14316  joinval  17609  meetval  17623  ipole  17762  sylow1  18722  frgpuplem  18892  uspgr2wlkeq  27421  wlkres  27446  wlkp1lem8  27456  usgr2pthlem  27538  2wlkdlem10  27708  1wlkdlem4  27913  3wlkdlem6  27938  3wlkdlem10  27942  pfxwlk  32365  oppr  43258  imarnf1pr  43474  elsprel  43630  sprsymrelf1lem  43646  sprsymrelf  43650  paireqne  43666  sbcpr  43676  isomuspgrlem2b  43987  isomuspgrlem2d  43989