Theorem breqtrrdi 4072

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 4071	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1364   class class class wbr 4030

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 3158  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031

This theorem is referenced by:  enpr2d  6873  fiunsnnn  6939  exmidpw2en  6970  unsnfi  6977  eninl  7158  eninr  7159  difinfinf  7162  exmidfodomrlemr  7264  exmidfodomrlemrALT  7265  dju1en  7275  djucomen  7278  djuassen  7279  xpdjuen  7280  gtndiv  9415  intqfrac2  10393  uzenom  10499  xrmaxiflemval  11396  ege2le3  11817  eirraplem  11923  pcprendvds  12431  pcpremul  12434  pcfaclem  12490  infpnlem2  12501  2strstr1g  12742  lmcn2  14459  dveflem  14905  tangtx  15014  ioocosf1o  15030  lgsdirprm  15191  sbthom  15586  nconstwlpolemgt0  15624