Theorem breqtrrdi 4075

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 4074	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1364   class class class wbr 4033

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 3628  df-pr 3629  df-op 3631  df-br 4034

This theorem is referenced by:  enpr2d  6876  fiunsnnn  6942  exmidpw2en  6973  unsnfi  6980  eninl  7163  eninr  7164  difinfinf  7167  exmidfodomrlemr  7269  exmidfodomrlemrALT  7270  dju1en  7280  djucomen  7283  djuassen  7284  xpdjuen  7285  gtndiv  9421  intqfrac2  10411  uzenom  10517  xrmaxiflemval  11415  ege2le3  11836  eirraplem  11942  bitsfzo  12119  pcprendvds  12459  pcpremul  12462  pcfaclem  12518  infpnlem2  12529  2strstr1g  12799  lmcn2  14516  dveflem  14962  tangtx  15074  ioocosf1o  15090  lgsdirprm  15275  sbthom  15670  nconstwlpolemgt0  15708