Theorem breqtrrdi 4167

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 4166	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1402   class class class wbr 4125

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 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 theorem 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 3711  df-pr 3712  df-op 3714  df-br 4126

This theorem is referenced by:  enpr2d  7101  fiunsnnn  7175  exmidpw2en  7209  unsnfi  7216  2omapfi  7310  eninl  7427  eninr  7428  difinfinf  7431  exmidfodomrlemr  7544  exmidfodomrlemrALT  7545  dju1en  7559  djucomen  7562  djuassen  7563  xpdjuen  7564  gtndiv  9720  intqfrac2  10734  uzenom  10840  xrmaxiflemval  11994  ege2le3  12416  eirraplem  12522  bitsfzo  12700  pcprendvds  13047  pcpremul  13050  pcfaclem  13106  infpnlem2  13117  2strstr1g  13453  lmcn2  15304  dveflem  15750  tangtx  15862  ioocosf1o  15878  lgsdirprm  16067  sbthom  16976  nconstwlpolemgt0  17019