Theorem breqtrrdi 4101

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 2211	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	breqtrdi 4100	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1373   class class class wbr 4059

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060

This theorem is referenced by:  enpr2d  6935  fiunsnnn  7004  exmidpw2en  7035  unsnfi  7042  eninl  7225  eninr  7226  difinfinf  7229  exmidfodomrlemr  7341  exmidfodomrlemrALT  7342  dju1en  7356  djucomen  7359  djuassen  7360  xpdjuen  7361  gtndiv  9503  intqfrac2  10501  uzenom  10607  xrmaxiflemval  11676  ege2le3  12097  eirraplem  12203  bitsfzo  12381  pcprendvds  12728  pcpremul  12731  pcfaclem  12787  infpnlem2  12798  2strstr1g  13069  lmcn2  14867  dveflem  15313  tangtx  15425  ioocosf1o  15441  lgsdirprm  15626  sbthom  16167  nconstwlpolemgt0  16205