Theorem eqbrtrrdi 4128

Description: A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)

Hypotheses

Ref	Expression
eqbrtrrdi.1	⊢ (𝜑 → 𝐵 = 𝐴)
eqbrtrrdi.2	⊢ 𝐵𝑅𝐶

Assertion

Ref	Expression
eqbrtrrdi	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem eqbrtrrdi

Step	Hyp	Ref	Expression
1		eqbrtrrdi.1	. . 3 ⊢ (𝜑 → 𝐵 = 𝐴)
2	1	eqcomd 2237	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		eqbrtrrdi.2	. 2 ⊢ 𝐵𝑅𝐶
4	2, 3	eqbrtrdi 4127	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: (None)