Theorem breqtrrdi 4125

Description: A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)

Hypotheses

Ref	Expression
breqtrrdi.1	⊢ (𝜑 → 𝐴𝑅𝐵)
breqtrrdi.2	⊢ 𝐶 = 𝐵

Assertion

Ref	Expression
breqtrrdi	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem breqtrrdi

Step	Hyp	Ref	Expression
1		breqtrrdi.1	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
2		breqtrrdi.2	. . 3 ⊢ 𝐶 = 𝐵
3	2	eqcomi 2233	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	breqtrdi 4124	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1395   class class class wbr 4083

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084

This theorem is referenced by:  enpr2d  6980  fiunsnnn  7051  exmidpw2en  7085  unsnfi  7092  eninl  7275  eninr  7276  difinfinf  7279  exmidfodomrlemr  7391  exmidfodomrlemrALT  7392  dju1en  7406  djucomen  7409  djuassen  7410  xpdjuen  7411  gtndiv  9553  intqfrac2  10553  uzenom  10659  xrmaxiflemval  11777  ege2le3  12198  eirraplem  12304  bitsfzo  12482  pcprendvds  12829  pcpremul  12832  pcfaclem  12888  infpnlem2  12899  2strstr1g  13171  lmcn2  14970  dveflem  15416  tangtx  15528  ioocosf1o  15544  lgsdirprm  15729  sbthom  16482  nconstwlpolemgt0  16520