Theorem breqtrrdi 4086

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 2209	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	breqtrdi 4085	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1373   class class class wbr 4044

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045

This theorem is referenced by:  enpr2d  6911  fiunsnnn  6978  exmidpw2en  7009  unsnfi  7016  eninl  7199  eninr  7200  difinfinf  7203  exmidfodomrlemr  7310  exmidfodomrlemrALT  7311  dju1en  7325  djucomen  7328  djuassen  7329  xpdjuen  7330  gtndiv  9468  intqfrac2  10464  uzenom  10570  xrmaxiflemval  11561  ege2le3  11982  eirraplem  12088  bitsfzo  12266  pcprendvds  12613  pcpremul  12616  pcfaclem  12672  infpnlem2  12683  2strstr1g  12954  lmcn2  14752  dveflem  15198  tangtx  15310  ioocosf1o  15326  lgsdirprm  15511  sbthom  15969  nconstwlpolemgt0  16007