Theorem qseq12d

Description: Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023)

	Ref	Expression
Hypotheses	qseq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	qseq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	qseq12d	⊢ ( 𝜑 → ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		qseq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		qseq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3		qseq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐷 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐷 ) )