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Theorem breq12 4690
 Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
Assertion
Ref Expression
breq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))

Proof of Theorem breq12
StepHypRef Expression
1 breq1 4688 . 2 (𝐴 = 𝐵 → (𝐴𝑅𝐶𝐵𝑅𝐶))
2 breq2 4689 . 2 (𝐶 = 𝐷 → (𝐵𝑅𝐶𝐵𝑅𝐷))
31, 2sylan9bb 736 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   class class class wbr 4685 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686 This theorem is referenced by:  breq12i  4694  breq12d  4698  breqan12d  4701  rbropapd  5044  posn  5221  dfrel4  5620  isopolem  6635  poxp  7334  soxp  7335  fnse  7339  ecopover  7894  canth2g  8155  infxpen  8875  sornom  9137  dcomex  9307  zorn2lem6  9361  brdom6disj  9392  fpwwe2  9503  rankcf  9637  ltresr  9999  ltxrlt  10146  wloglei  10598  ltxr  11987  xrltnr  11991  xrltnsym  12008  xrlttri  12010  xrlttr  12011  brfi1uzind  13318  brfi1indALT  13320  f1olecpbl  16234  isfull  16617  isfth  16621  prslem  16978  pslem  17253  dirtr  17283  xrsdsval  19838  dvcvx  23828  axcontlem9  25897  isrusgr  26513  wlk2f  26581  istrlson  26659  upgrwlkdvspth  26691  ispthson  26694  isspthson  26695  crctcshwlk  26770  crctcsh  26772  2pthon3v  26908  umgr2wlk  26914  0pthonv  27107  1pthon2v  27131  uhgr3cyclex  27160  2sqmo  29777  mclsppslem  31606  dfpo2  31771  fununiq  31793  slerec  32048  elfix2  32136  poimirlem10  33549  poimirlem11  33550  monotoddzzfi  37824  sprsymrelfolem2  42068  lindepsnlininds  42566
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