Theorem eqsstrd

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

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

Proof

Step	Hyp	Ref	Expression
1		eqsstrd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eqsstrd.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
3	1	sseq1d	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶 ) )
4	2 3	mpbird	⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 )