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Metamath Proof Explorer

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 ) )