Theorem opkeq2 4061

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 3801	. . 3 ⊢ (A = B → {C, A} = {C, B})
2	1	preq2d 3807	. 2 ⊢ (A = B → {{C}, {C, A}} = {{C}, {C, B}})
3		df-opk 4059	. 2 ⊢ ⟪C, A⟫ = {{C}, {C, A}}
4		df-opk 4059	. 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 3738  {cpr 3739  ⟪copk 4058

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-opk 4059

This theorem is referenced by:  opkeq12  4062  opkeq2i  4064  opkeq2d  4067  opkelcnvkg  4250  otkelins2kg  4254  otkelins3kg  4255  elimakg  4258  opkelcokg  4262  elp6  4264  opksnelsik  4266  sikexlem  4296  insklem  4305  dfnnc2  4396  ltfintri  4467  lenltfin  4470  tfinltfin  4502  sfinltfin  4536  setconslem6  4737