Theorem dmqseqeq1d

Description: Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 26-Sep-2021)

	Ref	Expression
Hypothesis	dmqseqeq1d.1	⊢ ( 𝜑 → 𝑅 = 𝑆 )
Assertion	dmqseqeq1d	⊢ ( 𝜑 → ( ( dom 𝑅 / 𝑅 ) = 𝐴 ↔ ( dom 𝑆 / 𝑆 ) = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		dmqseqeq1d.1	⊢ ( 𝜑 → 𝑅 = 𝑆 )
2		dmqseqeq1	⊢ ( 𝑅 = 𝑆 → ( ( dom 𝑅 / 𝑅 ) = 𝐴 ↔ ( dom 𝑆 / 𝑆 ) = 𝐴 ) )
3	1 2	syl	⊢ ( 𝜑 → ( ( dom 𝑅 / 𝑅 ) = 𝐴 ↔ ( dom 𝑆 / 𝑆 ) = 𝐴 ) )