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  6997  fiunsnnn  7070  exmidpw2en  7104  unsnfi  7111  eninl  7296  eninr  7297  difinfinf  7300  exmidfodomrlemr  7413  exmidfodomrlemrALT  7414  dju1en  7428  djucomen  7431  djuassen  7432  xpdjuen  7433  gtndiv  9575  intqfrac2  10582  uzenom  10688  xrmaxiflemval  11828  ege2le3  12250  eirraplem  12356  bitsfzo  12534  pcprendvds  12881  pcpremul  12884  pcfaclem  12940  infpnlem2  12951  2strstr1g  13223  lmcn2  15023  dveflem  15469  tangtx  15581  ioocosf1o  15597  lgsdirprm  15782  sbthom  16681  nconstwlpolemgt0  16720