Theorem ineq12d

Description: Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Hypotheses	ineq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	ineq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	ineq12d	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		ineq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		ineq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3		ineq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐷 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐷 ) )