Theorem breqtrrdi 4128

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 4127	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1395   class class class wbr 4086

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 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087

This theorem is referenced by:  enpr2d  6992  fiunsnnn  7063  exmidpw2en  7097  unsnfi  7104  eninl  7287  eninr  7288  difinfinf  7291  exmidfodomrlemr  7403  exmidfodomrlemrALT  7404  dju1en  7418  djucomen  7421  djuassen  7422  xpdjuen  7423  gtndiv  9565  intqfrac2  10571  uzenom  10677  xrmaxiflemval  11801  ege2le3  12222  eirraplem  12328  bitsfzo  12506  pcprendvds  12853  pcpremul  12856  pcfaclem  12912  infpnlem2  12923  2strstr1g  13195  lmcn2  14994  dveflem  15440  tangtx  15552  ioocosf1o  15568  lgsdirprm  15753  sbthom  16566  nconstwlpolemgt0  16604