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Theorem stcltr1i 32112
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 6911 . . . . . 6 (𝑥 = 𝐴 → ((𝑆𝑥) = 1 ↔ (𝑆𝐴) = 1))
54imbi1d 340 . . . . 5 (𝑥 = 𝐴 → (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) ↔ ((𝑆𝐴) = 1 → (𝑆𝑦) = 1)))
6 sseq1 4007 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
75, 6imbi12d 343 . . . 4 (𝑥 = 𝐴 → ((((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦) ↔ (((𝑆𝐴) = 1 → (𝑆𝑦) = 1) → 𝐴𝑦)))
8 fveqeq2 6911 . . . . . 6 (𝑦 = 𝐵 → ((𝑆𝑦) = 1 ↔ (𝑆𝐵) = 1))
98imbi2d 339 . . . . 5 (𝑦 = 𝐵 → (((𝑆𝐴) = 1 → (𝑆𝑦) = 1) ↔ ((𝑆𝐴) = 1 → (𝑆𝐵) = 1)))
10 sseq2 4008 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
119, 10imbi12d 343 . . . 4 (𝑦 = 𝐵 → ((((𝑆𝐴) = 1 → (𝑆𝑦) = 1) → 𝐴𝑦) ↔ (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵)))
127, 11rspc2v 3622 . . 3 ((𝐴C𝐵C ) → (∀𝑥C𝑦C (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦) → (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵)))
132, 3, 12mp2an 690 . 2 (∀𝑥C𝑦C (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦) → (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵))
141, 13simplbiim 503 1 (𝜑 → (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3058  wss 3949  cfv 6553  1c1 11149   C cch 30767  Statescst 30800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561
This theorem is referenced by:  stcltr2i  32113  stcltrlem2  32115
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