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Theorem opeq12 3781
Description: Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
Assertion
Ref Expression
opeq12 ((𝐴 = 𝐶𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)

Proof of Theorem opeq12
StepHypRef Expression
1 opeq1 3779 . 2 (𝐴 = 𝐶 → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐵⟩)
2 opeq2 3780 . 2 (𝐵 = 𝐷 → ⟨𝐶, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
31, 2sylan9eq 2230 1 ((𝐴 = 𝐶𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  cop 3596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602
This theorem is referenced by:  opeq12i  3784  opeq12d  3787  cbvopab  4075  opth  4238  copsex2t  4246  copsex2g  4247  relop  4778  funopg  5251  fsn  5689  fnressn  5703  cbvoprab12  5949  eqopi  6173  f1o2ndf1  6229  tposoprab  6281  brecop  6625  th3q  6640  ecovcom  6642  ecovicom  6643  ecovass  6644  ecoviass  6645  ecovdi  6646  ecovidi  6647  xpf1o  6844  1qec  7387  enq0sym  7431  addnq0mo  7446  mulnq0mo  7447  addnnnq0  7448  mulnnnq0  7449  distrnq0  7458  mulcomnq0  7459  addassnq0  7461  addsrmo  7742  mulsrmo  7743  addsrpr  7744  mulsrpr  7745  axcnre  7880  fsumcnv  11445  fprodcnv  11633  eucalgval2  12053
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