Metamath Proof Explorer


Theorem dmqseqeq1

Description: Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019)

Ref Expression
Assertion dmqseqeq1 ( 𝑅 = 𝑆 → ( ( dom 𝑅 / 𝑅 ) = 𝐴 ↔ ( dom 𝑆 / 𝑆 ) = 𝐴 ) )

Proof

Step Hyp Ref Expression
1 dmqseq ( 𝑅 = 𝑆 → ( dom 𝑅 / 𝑅 ) = ( dom 𝑆 / 𝑆 ) )
2 1 eqeq1d ( 𝑅 = 𝑆 → ( ( dom 𝑅 / 𝑅 ) = 𝐴 ↔ ( dom 𝑆 / 𝑆 ) = 𝐴 ) )