Theorem breqtrrdi 4157

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 4156	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1398   class class class wbr 4115

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 3701  df-pr 3702  df-op 3704  df-br 4116

This theorem is referenced by:  enpr2d  7079  fiunsnnn  7153  exmidpw2en  7187  unsnfi  7194  2omapfi  7286  eninl  7403  eninr  7404  difinfinf  7407  exmidfodomrlemr  7520  exmidfodomrlemrALT  7521  dju1en  7535  djucomen  7538  djuassen  7539  xpdjuen  7540  gtndiv  9696  intqfrac2  10710  uzenom  10816  xrmaxiflemval  11966  ege2le3  12388  eirraplem  12494  bitsfzo  12672  pcprendvds  13019  pcpremul  13022  pcfaclem  13078  infpnlem2  13089  2strstr1g  13425  lmcn2  15276  dveflem  15722  tangtx  15834  ioocosf1o  15850  lgsdirprm  16039  sbthom  16948  nconstwlpolemgt0  16991