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

Theorem opeq1d 3811
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 3805 . 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 1364  cop 3622
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-sn 3625  df-pr 3626  df-op 3628
This theorem is referenced by:  oteq1  3814  oteq2  3815  opth  4267  cbvoprab2  5992  djuf1olem  7114  dfplpq2  7416  ltexnqq  7470  nnanq0  7520  addpinq1  7526  prarloclemlo  7556  prarloclem3  7559  prarloclem5  7562  prsrriota  7850  caucvgsrlemfv  7853  caucvgsr  7864  pitonnlem2  7909  pitonn  7910  recidpirq  7920  ax1rid  7939  axrnegex  7941  nntopi  7956  axcaucvglemval  7959  fseq1m1p1  10164  frecuzrdglem  10485  frecuzrdgg  10490  frecuzrdgdomlem  10491  frecuzrdgfunlem  10493  frecuzrdgsuctlem  10497  fsum2dlemstep  11580  fprod2dlemstep  11768  ennnfonelemp1  12566  ennnfonelemnn0  12582  setscomd  12662  imasaddvallemg  12901
  Copyright terms: Public domain W3C validator