Theorem 3sstr4d

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995) (Proof shortened by Eric Schmidt, 26-Jan-2007)

	Ref	Expression
Hypotheses	3sstr4d.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	3sstr4d.2	⊢ ( 𝜑 → 𝐶 = 𝐴 )
	3sstr4d.3	⊢ ( 𝜑 → 𝐷 = 𝐵 )
Assertion	3sstr4d	⊢ ( 𝜑 → 𝐶 ⊆ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3sstr4d.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		3sstr4d.2	⊢ ( 𝜑 → 𝐶 = 𝐴 )
3		3sstr4d.3	⊢ ( 𝜑 → 𝐷 = 𝐵 )
4	2 3	sseq12d	⊢ ( 𝜑 → ( 𝐶 ⊆ 𝐷 ↔ 𝐴 ⊆ 𝐵 ) )
5	1 4	mpbird	⊢ ( 𝜑 → 𝐶 ⊆ 𝐷 )