Metamath Proof Explorer


Theorem eqvrelsymb

Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995) (Revised by Mario Carneiro, 12-Aug-2015) (Revised and distinct variable conditions removed by Peter Mazsa, 2-Jun-2019.)

Ref Expression
Hypothesis eqvrelsymb.1 ( 𝜑 → EqvRel 𝑅 )
Assertion eqvrelsymb ( 𝜑 → ( 𝐴 𝑅 𝐵𝐵 𝑅 𝐴 ) )

Proof

Step Hyp Ref Expression
1 eqvrelsymb.1 ( 𝜑 → EqvRel 𝑅 )
2 1 adantr ( ( 𝜑𝐴 𝑅 𝐵 ) → EqvRel 𝑅 )
3 simpr ( ( 𝜑𝐴 𝑅 𝐵 ) → 𝐴 𝑅 𝐵 )
4 2 3 eqvrelsym ( ( 𝜑𝐴 𝑅 𝐵 ) → 𝐵 𝑅 𝐴 )
5 1 adantr ( ( 𝜑𝐵 𝑅 𝐴 ) → EqvRel 𝑅 )
6 simpr ( ( 𝜑𝐵 𝑅 𝐴 ) → 𝐵 𝑅 𝐴 )
7 5 6 eqvrelsym ( ( 𝜑𝐵 𝑅 𝐴 ) → 𝐴 𝑅 𝐵 )
8 4 7 impbida ( 𝜑 → ( 𝐴 𝑅 𝐵𝐵 𝑅 𝐴 ) )