Theorem sseq2

Description: Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998)

	Ref	Expression
Assertion	sseq2	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eqss	⊢ ( 𝐴 = 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) )
2		sstr2	⊢ ( 𝐶 ⊆ 𝐴 → ( 𝐴 ⊆ 𝐵 → 𝐶 ⊆ 𝐵 ) )
3	2	com12	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ⊆ 𝐴 → 𝐶 ⊆ 𝐵 ) )
4		sstr2	⊢ ( 𝐶 ⊆ 𝐵 → ( 𝐵 ⊆ 𝐴 → 𝐶 ⊆ 𝐴 ) )
5	4	com12	⊢ ( 𝐵 ⊆ 𝐴 → ( 𝐶 ⊆ 𝐵 → 𝐶 ⊆ 𝐴 ) )
6	3 5	anbiim	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) → ( 𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵 ) )
7	1 6	sylbi	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵 ) )