Theorem stcltr1i 32697

Description: Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
stcltr1.1	⊢ (𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦)))
stcltr1.2	⊢ 𝐴 ∈ Cℋ
stcltr1.3	⊢ 𝐵 ∈ Cℋ

Assertion

Ref	Expression
stcltr1i	⊢ (𝜑 → (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem stcltr1i

Step	Hyp	Ref	Expression
1		stcltr1.1	. 2 ⊢ (𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦)))
2		stcltr1.2	. . 3 ⊢ 𝐴 ∈ Cℋ
3		stcltr1.3	. . 3 ⊢ 𝐵 ∈ Cℋ
4		fveqeq2 6894	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) = 1 ↔ (𝑆‘𝐴) = 1))
5	4	imbi1d 344	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) ↔ ((𝑆‘𝐴) = 1 → (𝑆‘𝑦) = 1)))
6		sseq1 3963	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑦))
7	5, 6	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝐴 → ((((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦) ↔ (((𝑆‘𝐴) = 1 → (𝑆‘𝑦) = 1) → 𝐴 ⊆ 𝑦)))
8		fveqeq2 6894	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝑦) = 1 ↔ (𝑆‘𝐵) = 1))
9	8	imbi2d 343	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑆‘𝐴) = 1 → (𝑆‘𝑦) = 1) ↔ ((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1)))
10		sseq2 3964	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵))
11	9, 10	imbi12d 347	. . . 4 ⊢ (𝑦 = 𝐵 → ((((𝑆‘𝐴) = 1 → (𝑆‘𝑦) = 1) → 𝐴 ⊆ 𝑦) ↔ (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵)))
12	7, 11	rspc2v 3594	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦) → (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵)))
13	2, 3, 12	mp2an 705	. 2 ⊢ (∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦) → (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵))
14	1, 13	simplbiim 514	1 ⊢ (𝜑 → (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ⊆ wss 3906  ‘cfv 6540  1c1 11116   C_ℋ cch 31352  Statescst 31385

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548

This theorem is used by:  stcltr2i  32698  stcltrlem2  32700