Theorem breqtrrdi 4135

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 4134	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1398   class class class wbr 4093

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094

This theorem is referenced by:  enpr2d  7040  fiunsnnn  7113  exmidpw2en  7147  unsnfi  7154  eninl  7356  eninr  7357  difinfinf  7360  exmidfodomrlemr  7473  exmidfodomrlemrALT  7474  dju1en  7488  djucomen  7491  djuassen  7492  xpdjuen  7493  gtndiv  9636  intqfrac2  10644  uzenom  10750  xrmaxiflemval  11890  ege2le3  12312  eirraplem  12418  bitsfzo  12596  pcprendvds  12943  pcpremul  12946  pcfaclem  13002  infpnlem2  13013  2strstr1g  13285  lmcn2  15091  dveflem  15537  tangtx  15649  ioocosf1o  15665  lgsdirprm  15853  sbthom  16754  nconstwlpolemgt0  16797