Theorem 3sstr3d

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000)

	Ref	Expression
Hypotheses	3sstr3d.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	3sstr3d.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
	3sstr3d.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
Assertion	3sstr3d	⊢ ( 𝜑 → 𝐶 ⊆ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3sstr3d.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		3sstr3d.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
3		3sstr3d.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
4	2 3	sseq12d	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷 ) )
5	1 4	mpbid	⊢ ( 𝜑 → 𝐶 ⊆ 𝐷 )