Theorem stcltr1i 32885

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 6899	. . . . . 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 3934	. . . . 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 6899	. . . . . 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 3935	. . . . 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 3586	. . 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 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3896  ‘cfv 6543  1c1 11208   C_ℋ cch 31540  Statescst 31573

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6497  df-fv 6551

This theorem is used by:  stcltr2i  32886  stcltrlem2  32888