Theorem opkeq1 4059

Description: Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.)

Assertion

Ref	Expression
opkeq1	⊢ (A = B → ⟪A, C⟫ = ⟪B, C⟫)

Proof of Theorem opkeq1

Step	Hyp	Ref	Expression
1		sneq 3744	. . 3 ⊢ (A = B → {A} = {B})
2		preq1 3799	. . 3 ⊢ (A = B → {A, C} = {B, C})
3	1, 2	preq12d 3807	. 2 ⊢ (A = B → {{A}, {A, C}} = {{B}, {B, C}})
4		df-opk 4058	. 2 ⊢ ⟪A, C⟫ = {{A}, {A, C}}
5		df-opk 4058	. 2 ⊢ ⟪B, C⟫ = {{B}, {B, C}}
6	3, 4, 5	3eqtr4g 2410	1 ⊢ (A = B → ⟪A, C⟫ = ⟪B, C⟫)

Syntax hints:   → wi 4   = wceq 1642  {csn 3737  {cpr 3738  ⟪copk 4057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-opk 4058

This theorem is referenced by:  opkeq12  4061  opkeq1i  4062  opkeq1d  4065  opkthg  4131  opkelcnvkg  4249  otkelins2kg  4253  otkelins3kg  4254  opkelcokg  4261  opksnelsik  4265  opkelimagekg  4271  elimaksn  4283  sikexlem  4295  dfimak2  4298  insklem  4304  setswith  4321  ndisjrelk  4323  dfpw2  4327  dfaddc2  4381  dfnnc2  4395  nnsucelrlem1  4424  leltfintr  4458  ltfinex  4464  ltfintrilem1  4465  ltfintri  4466  ssfin  4470  eqpwrelk  4478  eqpw1relk  4479  ncfinraiselem2  4480  ncfinlowerlem1  4482  eqtfinrelk  4486  evenfinex  4503  oddfinex  4504  evenodddisjlem1  4515  nnadjoinlem1  4519  nnpweqlem1  4522  srelk  4524  sfintfinlem1  4531  tfinnnlem1  4533  sfinltfin  4535  spfinex  4537  vfinncvntnn  4548  vfinspss  4551  vfinncsp  4554  dfop2lem1  4573  setconslem1  4731  setconslem2  4732  setconslem3  4733  setconslem4  4734  setconslem6  4736  setconslem7  4737  df1st2  4738  dfswap2  4741