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 𝐵 ↔ 𝐵 𝑅 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvrelsymb.1 | ⊢ ( 𝜑 → EqvRel 𝑅 ) | |
2 | 1 | adantr | ⊢ ( ( 𝜑 ∧ 𝐴 𝑅 𝐵 ) → EqvRel 𝑅 ) |
3 | simpr | ⊢ ( ( 𝜑 ∧ 𝐴 𝑅 𝐵 ) → 𝐴 𝑅 𝐵 ) | |
4 | 2 3 | eqvrelsym | ⊢ ( ( 𝜑 ∧ 𝐴 𝑅 𝐵 ) → 𝐵 𝑅 𝐴 ) |
5 | 1 | adantr | ⊢ ( ( 𝜑 ∧ 𝐵 𝑅 𝐴 ) → EqvRel 𝑅 ) |
6 | simpr | ⊢ ( ( 𝜑 ∧ 𝐵 𝑅 𝐴 ) → 𝐵 𝑅 𝐴 ) | |
7 | 5 6 | eqvrelsym | ⊢ ( ( 𝜑 ∧ 𝐵 𝑅 𝐴 ) → 𝐴 𝑅 𝐵 ) |
8 | 4 7 | impbida | ⊢ ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ 𝐵 𝑅 𝐴 ) ) |