Metamath Proof Explorer


Theorem dfeqvrel2

Description: Alternate definition of the equivalence relation predicate. (Contributed by Peter Mazsa, 22-Apr-2019)

Ref Expression
Assertion dfeqvrel2 EqvRelRIdomRRR-1RRRRRelR

Proof

Step Hyp Ref Expression
1 df-eqvrel EqvRelRRefRelRSymRelRTrRelR
2 refsymrel2 RefRelRSymRelRIdomRRR-1RRelR
3 dftrrel2 TrRelRRRRRelR
4 2 3 anbi12i RefRelRSymRelRTrRelRIdomRRR-1RRelRRRRRelR
5 df-3an RefRelRSymRelRTrRelRRefRelRSymRelRTrRelR
6 df-3an IdomRRR-1RRRRIdomRRR-1RRRR
7 6 anbi1i IdomRRR-1RRRRRelRIdomRRR-1RRRRRelR
8 3anan32 IdomRRR-1RRelRRRRIdomRRR-1RRRRRelR
9 anandi3r IdomRRR-1RRelRRRRIdomRRR-1RRelRRRRRelR
10 7 8 9 3bitr2i IdomRRR-1RRRRRelRIdomRRR-1RRelRRRRRelR
11 4 5 10 3bitr4i RefRelRSymRelRTrRelRIdomRRR-1RRRRRelR
12 1 11 bitri EqvRelRIdomRRR-1RRRRRelR