Theorem opkeq2 4060

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

Assertion

Ref	Expression
opkeq2	⊢ (A = B → ⟪C, A⟫ = ⟪C, B⟫)

Proof of Theorem opkeq2

Step	Hyp	Ref	Expression
1		preq2 3800	. . 3 ⊢ (A = B → {C, A} = {C, B})
2	1	preq2d 3806	. 2 ⊢ (A = B → {{C}, {C, A}} = {{C}, {C, B}})
3		df-opk 4058	. 2 ⊢ ⟪C, A⟫ = {{C}, {C, A}}
4		df-opk 4058	. 2 ⊢ ⟪C, B⟫ = {{C}, {C, B}}
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → ⟪C, A⟫ = ⟪C, B⟫)

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  opkeq2i  4063  opkeq2d  4066  opkelcnvkg  4249  otkelins2kg  4253  otkelins3kg  4254  elimakg  4257  opkelcokg  4261  elp6  4263  opksnelsik  4265  sikexlem  4295  insklem  4304  dfnnc2  4395  ltfintri  4466  lenltfin  4469  tfinltfin  4501  sfinltfin  4535  setconslem6  4736