Theorem dmqseqeq1d

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

	Ref	Expression
Hypothesis	dmqseqeq1d.1	|- ( ph -> R = S )
Assertion	dmqseqeq1d	|- ( ph -> ( ( dom R /. R ) = A <-> ( dom S /. S ) = A ) )

Proof

Step	Hyp	Ref	Expression
1		dmqseqeq1d.1	|- ( ph -> R = S )
2		dmqseqeq1	|- ( R = S -> ( ( dom R /. R ) = A <-> ( dom S /. S ) = A ) )
3	1 2	syl	|- ( ph -> ( ( dom R /. R ) = A <-> ( dom S /. S ) = A ) )