Theorem breqtrrdi 4130

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 2235	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	breqtrdi 4129	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1397   class class class wbr 4088

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089

This theorem is referenced by:  enpr2d  6996  fiunsnnn  7069  exmidpw2en  7103  unsnfi  7110  eninl  7295  eninr  7296  difinfinf  7299  exmidfodomrlemr  7412  exmidfodomrlemrALT  7413  dju1en  7427  djucomen  7430  djuassen  7431  xpdjuen  7432  gtndiv  9574  intqfrac2  10580  uzenom  10686  xrmaxiflemval  11810  ege2le3  12231  eirraplem  12337  bitsfzo  12515  pcprendvds  12862  pcpremul  12865  pcfaclem  12921  infpnlem2  12932  2strstr1g  13204  lmcn2  15003  dveflem  15449  tangtx  15561  ioocosf1o  15577  lgsdirprm  15762  sbthom  16630  nconstwlpolemgt0  16668