Theorem erALTVeq1

Description: Equality theorem for equivalence relation on domain quotient. (Contributed by Peter Mazsa, 25-Sep-2021)

	Ref	Expression
Assertion	erALTVeq1	⊢ ( 𝑅 = 𝑆 → ( 𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		eqvreleq	⊢ ( 𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆 ) )
2		dmqseqeq1	⊢ ( 𝑅 = 𝑆 → ( ( dom 𝑅 / 𝑅 ) = 𝐴 ↔ ( dom 𝑆 / 𝑆 ) = 𝐴 ) )
3	1 2	anbi12d	⊢ ( 𝑅 = 𝑆 → ( ( EqvRel 𝑅 ∧ ( dom 𝑅 / 𝑅 ) = 𝐴 ) ↔ ( EqvRel 𝑆 ∧ ( dom 𝑆 / 𝑆 ) = 𝐴 ) ) )
4		dferALTV2	⊢ ( 𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ ( dom 𝑅 / 𝑅 ) = 𝐴 ) )
5		dferALTV2	⊢ ( 𝑆 ErALTV 𝐴 ↔ ( EqvRel 𝑆 ∧ ( dom 𝑆 / 𝑆 ) = 𝐴 ) )
6	3 4 5	3bitr4g	⊢ ( 𝑅 = 𝑆 → ( 𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴 ) )