Theorem breqtrrdi 4172

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 2242	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	breqtrdi 4171	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   class class class wbr 4130

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131

This theorem is used by:  enpr2d  7111  fiunsnnn  7185  exmidpw2en  7219  unsnfi  7226  2omapfi  7320  eninl  7437  eninr  7438  difinfinf  7441  exmidfodomrlemr  7554  exmidfodomrlemrALT  7555  dju1en  7569  djucomen  7572  djuassen  7573  xpdjuen  7574  gtndiv  9741  intqfrac2  10756  uzenom  10862  xrmaxiflemval  12016  ege2le3  12438  eirraplem  12544  bitsfzo  12722  pcprendvds  13069  pcpremul  13072  pcfaclem  13128  infpnlem2  13139  2strstr1g  13476  lmcn2  15381  dveflem  15827  tangtx  15939  ioocosf1o  15955  lgsdirprm  16153  sbthom  17071  nconstwlpolemgt0  17114