Theorem sseq12i

Description: An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999) (Proof shortened by Eric Schmidt, 26-Jan-2007)

	Ref	Expression
Hypotheses	sseq1i.1	⊢ 𝐴 = 𝐵
	sseq12i.2	⊢ 𝐶 = 𝐷
Assertion	sseq12i	⊢ ( 𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		sseq1i.1	⊢ 𝐴 = 𝐵
2		sseq12i.2	⊢ 𝐶 = 𝐷
3		sseq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷 ) )
4	1 2 3	mp2an	⊢ ( 𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷 )