Theorem eqbrtrrdi 4165

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 2244	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		eqbrtrrdi.2	. 2 ⊢ 𝐵𝑅𝐶
4	2, 3	eqbrtrdi 4164	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:  ballotfilemic  13228  ballotfilem1c  13229