ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opeq1d GIF version

Theorem opeq1d 3679
Description: Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
Hypothesis
Ref Expression
opeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
opeq1d (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Proof of Theorem opeq1d
StepHypRef Expression
1 opeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 opeq1 3673 . 2 (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
31, 2syl 14 1 (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  cop 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-op 3504
This theorem is referenced by:  oteq1  3682  oteq2  3683  opth  4127  cbvoprab2  5810  djuf1olem  6904  dfplpq2  7126  ltexnqq  7180  nnanq0  7230  addpinq1  7236  prarloclemlo  7266  prarloclem3  7269  prarloclem5  7272  prsrriota  7560  caucvgsrlemfv  7563  caucvgsr  7574  pitonnlem2  7619  pitonn  7620  recidpirq  7630  ax1rid  7649  axrnegex  7651  nntopi  7666  axcaucvglemval  7669  fseq1m1p1  9826  frecuzrdglem  10135  frecuzrdgg  10140  frecuzrdgdomlem  10141  frecuzrdgfunlem  10143  frecuzrdgsuctlem  10147  fsum2dlemstep  11154  ennnfonelemp1  11825  ennnfonelemnn0  11841
  Copyright terms: Public domain W3C validator