Theorem breqtrrdi 4076

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 2200	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	breqtrdi 4075	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1364   class class class wbr 4034

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035

This theorem is referenced by:  enpr2d  6885  fiunsnnn  6951  exmidpw2en  6982  unsnfi  6989  eninl  7172  eninr  7173  difinfinf  7176  exmidfodomrlemr  7281  exmidfodomrlemrALT  7282  dju1en  7296  djucomen  7299  djuassen  7300  xpdjuen  7301  gtndiv  9438  intqfrac2  10428  uzenom  10534  xrmaxiflemval  11432  ege2le3  11853  eirraplem  11959  bitsfzo  12137  pcprendvds  12484  pcpremul  12487  pcfaclem  12543  infpnlem2  12554  2strstr1g  12824  lmcn2  14600  dveflem  15046  tangtx  15158  ioocosf1o  15174  lgsdirprm  15359  sbthom  15757  nconstwlpolemgt0  15795