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Theorem stcltr1i 30060
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 ∧ ∀𝑥C𝑦C (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦)))
stcltr1.2 𝐴C
stcltr1.3 𝐵C
Assertion
Ref Expression
stcltr1i (𝜑 → (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem stcltr1i
StepHypRef Expression
1 stcltr1.1 . 2 (𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥C𝑦C (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦)))
2 stcltr1.2 . . 3 𝐴C
3 stcltr1.3 . . 3 𝐵C
4 fveqeq2 6658 . . . . . 6 (𝑥 = 𝐴 → ((𝑆𝑥) = 1 ↔ (𝑆𝐴) = 1))
54imbi1d 345 . . . . 5 (𝑥 = 𝐴 → (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) ↔ ((𝑆𝐴) = 1 → (𝑆𝑦) = 1)))
6 sseq1 3943 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
75, 6imbi12d 348 . . . 4 (𝑥 = 𝐴 → ((((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦) ↔ (((𝑆𝐴) = 1 → (𝑆𝑦) = 1) → 𝐴𝑦)))
8 fveqeq2 6658 . . . . . 6 (𝑦 = 𝐵 → ((𝑆𝑦) = 1 ↔ (𝑆𝐵) = 1))
98imbi2d 344 . . . . 5 (𝑦 = 𝐵 → (((𝑆𝐴) = 1 → (𝑆𝑦) = 1) ↔ ((𝑆𝐴) = 1 → (𝑆𝐵) = 1)))
10 sseq2 3944 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
119, 10imbi12d 348 . . . 4 (𝑦 = 𝐵 → ((((𝑆𝐴) = 1 → (𝑆𝑦) = 1) → 𝐴𝑦) ↔ (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵)))
127, 11rspc2v 3584 . . 3 ((𝐴C𝐵C ) → (∀𝑥C𝑦C (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦) → (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵)))
132, 3, 12mp2an 691 . 2 (∀𝑥C𝑦C (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦) → (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵))
141, 13simplbiim 508 1 (𝜑 → (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  wss 3884  cfv 6328  1c1 10531   C cch 28715  Statescst 28748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336
This theorem is referenced by:  stcltr2i  30061  stcltrlem2  30063
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