Theorem breqtrrdi 4071

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 2197	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	breqtrdi 4070	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1364   class class class wbr 4029

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030

This theorem is referenced by:  enpr2d  6871  fiunsnnn  6937  exmidpw2en  6968  unsnfi  6975  eninl  7156  eninr  7157  difinfinf  7160  exmidfodomrlemr  7262  exmidfodomrlemrALT  7263  dju1en  7273  djucomen  7276  djuassen  7277  xpdjuen  7278  gtndiv  9412  intqfrac2  10390  uzenom  10496  xrmaxiflemval  11393  ege2le3  11814  eirraplem  11920  pcprendvds  12428  pcpremul  12431  pcfaclem  12487  infpnlem2  12498  2strstr1g  12739  lmcn2  14448  dveflem  14872  tangtx  14973  ioocosf1o  14989  lgsdirprm  15150  sbthom  15516  nconstwlpolemgt0  15554