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Metamath Proof Explorer

Theorem eqvrelsym

Description: An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995) (Revised by Mario Carneiro, 12-Aug-2015) (Revised by Peter Mazsa, 2-Jun-2019)

		Ref	Expression
	Hypotheses	eqvrelsym.1	⊢ ( 𝜑 → EqvRel 𝑅 )
		eqvrelsym.2	⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
	Assertion	eqvrelsym	⊢ ( 𝜑 → 𝐵 𝑅 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		eqvrelsym.1	⊢ ( 𝜑 → EqvRel 𝑅 )
2		eqvrelsym.2	⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
3		eqvrelrel	⊢ ( EqvRel 𝑅 → Rel 𝑅 )
4		relbrcnvg	⊢ ( Rel 𝑅 → ( 𝐵 ^◡ 𝑅 𝐴 ↔ 𝐴 𝑅 𝐵 ) )
5	1 3 4	3syl	⊢ ( 𝜑 → ( 𝐵 ^◡ 𝑅 𝐴 ↔ 𝐴 𝑅 𝐵 ) )
6	2 5	mpbird	⊢ ( 𝜑 → 𝐵 ^◡ 𝑅 𝐴 )
7		eqvrelsymrel	⊢ ( EqvRel 𝑅 → SymRel 𝑅 )
8		dfsymrel2	⊢ ( SymRel 𝑅 ↔ ( ^◡ 𝑅 ⊆ 𝑅 ∧ Rel 𝑅 ) )
9	8	simplbi	⊢ ( SymRel 𝑅 → ^◡ 𝑅 ⊆ 𝑅 )
10	1 7 9	3syl	⊢ ( 𝜑 → ^◡ 𝑅 ⊆ 𝑅 )
11	10	ssbrd	⊢ ( 𝜑 → ( 𝐵 ^◡ 𝑅 𝐴 → 𝐵 𝑅 𝐴 ) )
12	6 11	mpd	⊢ ( 𝜑 → 𝐵 𝑅 𝐴 )