Theorem breqtrrdi 4047

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 2181	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	breqtrdi 4046	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1353   class class class wbr 4005

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006

This theorem is referenced by:  enpr2d  6819  fiunsnnn  6883  unsnfi  6920  eninl  7098  eninr  7099  difinfinf  7102  exmidfodomrlemr  7203  exmidfodomrlemrALT  7204  dju1en  7214  djucomen  7217  djuassen  7218  xpdjuen  7219  gtndiv  9350  intqfrac2  10321  uzenom  10427  xrmaxiflemval  11260  ege2le3  11681  eirraplem  11786  pcprendvds  12292  pcpremul  12295  pcfaclem  12349  infpnlem2  12360  2strstr1g  12582  lmcn2  13865  dveflem  14272  tangtx  14344  ioocosf1o  14360  lgsdirprm  14520  sbthom  14859  nconstwlpolemgt0  14897