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  6972  fiunsnnn  7043  exmidpw2en  7074  unsnfi  7081  eninl  7264  eninr  7265  difinfinf  7268  exmidfodomrlemr  7380  exmidfodomrlemrALT  7381  dju1en  7395  djucomen  7398  djuassen  7399  xpdjuen  7400  gtndiv  9542  intqfrac2  10541  uzenom  10647  xrmaxiflemval  11761  ege2le3  12182  eirraplem  12288  bitsfzo  12466  pcprendvds  12813  pcpremul  12816  pcfaclem  12872  infpnlem2  12883  2strstr1g  13155  lmcn2  14954  dveflem  15400  tangtx  15512  ioocosf1o  15528  lgsdirprm  15713  sbthom  16394  nconstwlpolemgt0  16432