Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  breqan12d GIF version

Theorem breqan12d 3945
 Description: Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
Hypotheses
Ref Expression
breq1d.1 (𝜑𝐴 = 𝐵)
breqan12i.2 (𝜓𝐶 = 𝐷)
Assertion
Ref Expression
breqan12d ((𝜑𝜓) → (𝐴𝑅𝐶𝐵𝑅𝐷))

Proof of Theorem breqan12d
StepHypRef Expression
1 breq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 breqan12i.2 . 2 (𝜓𝐶 = 𝐷)
3 breq12 3934 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))
41, 2, 3syl2an 287 1 ((𝜑𝜓) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   class class class wbr 3929 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930 This theorem is referenced by:  breqan12rd  3946  sosng  4612  isoresbr  5710  isoid  5711  isores3  5716  isoini2  5720  ofrfval  5990  oviec  6535  enqbreq2  7172  ltresr2  7655  axpre-ltadd  7701  leltadd  8216  xltneg  9626  lt2sq  10373  le2sq  10374  cnreim  10757  sqrtle  10815  sqrtlt  10816  absext  10842  reefiso  12875  logltb  12969
 Copyright terms: Public domain W3C validator