Theorem disjdmqseqeq1

Description: Lemma for the equality theorem for partition ~? parteq1 . (Contributed by Peter Mazsa, 5-Oct-2021)

	Ref	Expression
Assertion	disjdmqseqeq1	⊢ ( 𝑅 = 𝑆 → ( ( Disj 𝑅 ∧ ( dom 𝑅 / 𝑅 ) = 𝐴 ) ↔ ( Disj 𝑆 ∧ ( dom 𝑆 / 𝑆 ) = 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		disjeq	⊢ ( 𝑅 = 𝑆 → ( Disj 𝑅 ↔ Disj 𝑆 ) )
2		dmqseqeq1	⊢ ( 𝑅 = 𝑆 → ( ( dom 𝑅 / 𝑅 ) = 𝐴 ↔ ( dom 𝑆 / 𝑆 ) = 𝐴 ) )
3	1 2	anbi12d	⊢ ( 𝑅 = 𝑆 → ( ( Disj 𝑅 ∧ ( dom 𝑅 / 𝑅 ) = 𝐴 ) ↔ ( Disj 𝑆 ∧ ( dom 𝑆 / 𝑆 ) = 𝐴 ) ) )