Theorem opeq12 4379

Description: Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)

Assertion

Ref	Expression
opeq12	⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)

Proof of Theorem opeq12

Step	Hyp	Ref	Expression
1		opeq1 4377	. 2 ⊢ (𝐴 = 𝐶 → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐵⟩)
2		opeq2 4378	. 2 ⊢ (𝐵 = 𝐷 → ⟨𝐶, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
3	1, 2	sylan9eq 2675	1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480  ⟨cop 4161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162

This theorem is referenced by:  opeq12i  4382  opeq12d  4385  cbvopab  4693  opth  4915  copsex2t  4927  copsex2g  4928  relop  5242  funopg  5890  fvn0ssdmfun  6316  fsn  6367  fnressn  6390  fmptsng  6399  fmptsnd  6400  tpres  6431  cbvoprab12  6694  eqopi  7162  f1o2ndf1  7245  tposoprab  7348  omeu  7625  brecop  7800  ecovcom  7814  ecovass  7815  ecovdi  7816  xpf1o  8082  addsrmo  9854  mulsrmo  9855  addsrpr  9856  mulsrpr  9857  addcnsr  9916  axcnre  9945  seqeq1  12760  opfi1uzind  13238  opfi1uzindOLD  13244  fsumcnv  14451  fprodcnv  14657  eucalgval2  15237  xpstopnlem1  21552  qustgplem  21864  brabgaf  29304  qqhval2  29850  brsegle  31910  finxpreclem3  32901  dvnprodlem1  39498  funop1  40629  uspgrsprf1  41073