Theorem breqtrrdi 4151

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 2236	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	breqtrdi 4150	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1398   class class class wbr 4109

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110

This theorem is referenced by:  enpr2d  7064  fiunsnnn  7138  exmidpw2en  7172  unsnfi  7179  2omapfi  7271  eninl  7388  eninr  7389  difinfinf  7392  exmidfodomrlemr  7505  exmidfodomrlemrALT  7506  dju1en  7520  djucomen  7523  djuassen  7524  xpdjuen  7525  gtndiv  9673  intqfrac2  10681  uzenom  10787  xrmaxiflemval  11935  ege2le3  12357  eirraplem  12463  bitsfzo  12641  pcprendvds  12988  pcpremul  12991  pcfaclem  13047  infpnlem2  13058  2strstr1g  13335  lmcn2  15145  dveflem  15591  tangtx  15703  ioocosf1o  15719  lgsdirprm  15907  sbthom  16806  nconstwlpolemgt0  16850