Theorem breqtrrid 4126

Description: B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)

Hypotheses

Ref	Expression
breqtrrid.1	⊢ 𝐴𝑅𝐵
breqtrrid.2	⊢ (𝜑 → 𝐶 = 𝐵)

Assertion

Ref	Expression
breqtrrid	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem breqtrrid

Step	Hyp	Ref	Expression
1		breqtrrid.1	. 2 ⊢ 𝐴𝑅𝐵
2		breqtrrid.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐵)
3	2	eqcomd 2237	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	1, 3	breqtrid 4125	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1397   class class class wbr 4088

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089

This theorem is referenced by:  xsubge0  10116  xposdif  10117  bernneq  10923  bitsfzo  12518  bitsmod  12519  bitsinv1lem  12524  pcge0  12888  rpabscxpbnd  15667  lgsdir2lem2  15761  2lgsoddprmlem3  15843  eupth2lem3lem3fi  16324  eupth2lembfi  16331  trilpolemclim  16661  trilpolemlt1  16666  nconstwlpolemgt0  16689