Theorem breqtrrdi 4156

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 2238	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	breqtrdi 4155	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1398   class class class wbr 4114

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 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115

This theorem is referenced by:  enpr2d  7077  fiunsnnn  7151  exmidpw2en  7185  unsnfi  7192  2omapfi  7284  eninl  7401  eninr  7402  difinfinf  7405  exmidfodomrlemr  7518  exmidfodomrlemrALT  7519  dju1en  7533  djucomen  7536  djuassen  7537  xpdjuen  7538  gtndiv  9691  intqfrac2  10705  uzenom  10811  xrmaxiflemval  11960  ege2le3  12382  eirraplem  12488  bitsfzo  12666  pcprendvds  13013  pcpremul  13016  pcfaclem  13072  infpnlem2  13083  2strstr1g  13419  lmcn2  15271  dveflem  15717  tangtx  15829  ioocosf1o  15845  lgsdirprm  16033  sbthom  16932  nconstwlpolemgt0  16976