Theorem strb

Description: Strong state theorem (bidirectional version). (Contributed by NM, 7-Apr-2001) (New usage is discouraged.)

	Ref	Expression
Hypotheses	str.1	⊢ 𝐴 ∈ Cℋ
	str.2	⊢ 𝐵 ∈ Cℋ
Assertion	strb	⊢ ( ∀ 𝑓 ∈ States ( ( 𝑓 ‘ 𝐴 ) = 1 → ( 𝑓 ‘ 𝐵 ) = 1 ) ↔ 𝐴 ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		str.1	⊢ 𝐴 ∈ Cℋ
2		str.2	⊢ 𝐵 ∈ Cℋ
3	1 2	stri	⊢ ( ∀ 𝑓 ∈ States ( ( 𝑓 ‘ 𝐴 ) = 1 → ( 𝑓 ‘ 𝐵 ) = 1 ) → 𝐴 ⊆ 𝐵 )
4	1 2	stlesi	⊢ ( 𝑓 ∈ States → ( 𝐴 ⊆ 𝐵 → ( ( 𝑓 ‘ 𝐴 ) = 1 → ( 𝑓 ‘ 𝐵 ) = 1 ) ) )
5	4	com12	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑓 ∈ States → ( ( 𝑓 ‘ 𝐴 ) = 1 → ( 𝑓 ‘ 𝐵 ) = 1 ) ) )
6	5	ralrimiv	⊢ ( 𝐴 ⊆ 𝐵 → ∀ 𝑓 ∈ States ( ( 𝑓 ‘ 𝐴 ) = 1 → ( 𝑓 ‘ 𝐵 ) = 1 ) )
7	3 6	impbii	⊢ ( ∀ 𝑓 ∈ States ( ( 𝑓 ‘ 𝐴 ) = 1 → ( 𝑓 ‘ 𝐵 ) = 1 ) ↔ 𝐴 ⊆ 𝐵 )