Theorem preq12 4679

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 4677	. 2 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
2		preq2 4678	. 2 ⊢ (𝐵 = 𝐷 → {𝐶, 𝐵} = {𝐶, 𝐷})
3	1, 2	sylan9eq 2791	1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  {cpr 4569

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 2708

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 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-un 3894  df-sn 4568  df-pr 4570

This theorem is referenced by:  preq12i  4682  preq12d  4685  ssprsseq  4768  preq12b  4793  prnebg  4799  preq12nebg  4806  opthprneg  4808  elpr2elpr  4812  relop  5805  opthreg  9539  hashle2pr  14439  wwlktovfo  14920  joinval  18341  meetval  18355  ipole  18500  sylow1  19578  frgpuplem  19747  uspgr2wlkeq  29714  wlkres  29737  wlkp1lem8  29747  usgr2pthlem  29831  2wlkdlem10  30003  1wlkdlem4  30210  3wlkdlem6  30235  3wlkdlem10  30239  pfxwlk  35306  oppr  47478  imarnf1pr  47730  elsprel  47935  sprsymrelf1lem  47951  sprsymrelf  47955  paireqne  47971  sbcpr  47981  isuspgrimlem  48371  grtrif1o  48418