Theorem qseq12

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

	Ref	Expression
Assertion	qseq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		qseq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) )
2		qseq2	⊢ ( 𝐶 = 𝐷 → ( 𝐵 / 𝐶 ) = ( 𝐵 / 𝐷 ) )
3	1 2	sylan9eq	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐷 ) )