Theorem breqtrrdi 4031

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 2174	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	breqtrdi 4030	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1348   class class class wbr 3989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990

This theorem is referenced by:  enpr2d  6795  fiunsnnn  6859  unsnfi  6896  eninl  7074  eninr  7075  difinfinf  7078  exmidfodomrlemr  7179  exmidfodomrlemrALT  7180  dju1en  7190  djucomen  7193  djuassen  7194  xpdjuen  7195  gtndiv  9307  intqfrac2  10275  uzenom  10381  xrmaxiflemval  11213  ege2le3  11634  eirraplem  11739  pcprendvds  12244  pcpremul  12247  pcfaclem  12301  infpnlem2  12312  2strstr1g  12521  lmcn2  13074  dveflem  13481  tangtx  13553  ioocosf1o  13569  lgsdirprm  13729  sbthom  14058  nconstwlpolemgt0  14095