Theorem breqtrrdi 4045

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 2181	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	breqtrdi 4044	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1353   class class class wbr 4003

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004

This theorem is referenced by:  enpr2d  6816  fiunsnnn  6880  unsnfi  6917  eninl  7095  eninr  7096  difinfinf  7099  exmidfodomrlemr  7200  exmidfodomrlemrALT  7201  dju1en  7211  djucomen  7214  djuassen  7215  xpdjuen  7216  gtndiv  9347  intqfrac2  10318  uzenom  10424  xrmaxiflemval  11257  ege2le3  11678  eirraplem  11783  pcprendvds  12289  pcpremul  12292  pcfaclem  12346  infpnlem2  12357  2strstr1g  12579  lmcn2  13750  dveflem  14157  tangtx  14229  ioocosf1o  14245  lgsdirprm  14405  sbthom  14744  nconstwlpolemgt0  14781