Theorem sseqtrd

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004)

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

Proof

Step	Hyp	Ref	Expression
1		sseqtrd.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		sseqtrd.2	⊢ ( 𝜑 → 𝐵 = 𝐶 )
3	2	sseq2d	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶 ) )
4	1 3	mpbid	⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 )