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  6878  fiunsnnn  6944  exmidpw2en  6975  unsnfi  6982  eninl  7165  eninr  7166  difinfinf  7169  exmidfodomrlemr  7272  exmidfodomrlemrALT  7273  dju1en  7283  djucomen  7286  djuassen  7287  xpdjuen  7288  gtndiv  9424  intqfrac2  10414  uzenom  10520  xrmaxiflemval  11418  ege2le3  11839  eirraplem  11945  bitsfzo  12123  pcprendvds  12470  pcpremul  12473  pcfaclem  12529  infpnlem2  12540  2strstr1g  12810  lmcn2  14542  dveflem  14988  tangtx  15100  ioocosf1o  15116  lgsdirprm  15301  sbthom  15697  nconstwlpolemgt0  15735