Theorem 3sstr3g

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

	Ref	Expression
Hypotheses	3sstr3g.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	3sstr3g.2	⊢ 𝐴 = 𝐶
	3sstr3g.3	⊢ 𝐵 = 𝐷
Assertion	3sstr3g	⊢ ( 𝜑 → 𝐶 ⊆ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3sstr3g.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		3sstr3g.2	⊢ 𝐴 = 𝐶
3		3sstr3g.3	⊢ 𝐵 = 𝐷
4	2 3	sseq12i	⊢ ( 𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷 )
5	1 4	sylib	⊢ ( 𝜑 → 𝐶 ⊆ 𝐷 )