Theorem ineqan12d

Description: Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007)

	Ref	Expression
Hypotheses	ineq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	ineqan12d.2	⊢ ( 𝜓 → 𝐶 = 𝐷 )
Assertion	ineqan12d	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		ineq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		ineqan12d.2	⊢ ( 𝜓 → 𝐶 = 𝐷 )
3		ineq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐷 ) )
4	1 2 3	syl2an	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐷 ) )