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		sstr2	⊢ ( 𝐶 ⊆ 𝐴 → ( 𝐴 ⊆ 𝐵 → 𝐶 ⊆ 𝐵 ) )
2	1	com12	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ⊆ 𝐴 → 𝐶 ⊆ 𝐵 ) )
3		sstr2	⊢ ( 𝐶 ⊆ 𝐵 → ( 𝐵 ⊆ 𝐴 → 𝐶 ⊆ 𝐴 ) )
4	3	com12	⊢ ( 𝐵 ⊆ 𝐴 → ( 𝐶 ⊆ 𝐵 → 𝐶 ⊆ 𝐴 ) )
5	2 4	anim12i	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) → ( ( 𝐶 ⊆ 𝐴 → 𝐶 ⊆ 𝐵 ) ∧ ( 𝐶 ⊆ 𝐵 → 𝐶 ⊆ 𝐴 ) ) )
6		eqss	⊢ ( 𝐴 = 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) )
7		dfbi2	⊢ ( ( 𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵 ) ↔ ( ( 𝐶 ⊆ 𝐴 → 𝐶 ⊆ 𝐵 ) ∧ ( 𝐶 ⊆ 𝐵 → 𝐶 ⊆ 𝐴 ) ) )
8	5 6 7	3imtr4i	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵 ) )