Theorem sseqtrrd

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

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

Proof

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