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Theorem stcltr1i 31214
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 6851 . . . . . 6 (𝑥 = 𝐴 → ((𝑆𝑥) = 1 ↔ (𝑆𝐴) = 1))
54imbi1d 341 . . . . 5 (𝑥 = 𝐴 → (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) ↔ ((𝑆𝐴) = 1 → (𝑆𝑦) = 1)))
6 sseq1 3969 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
75, 6imbi12d 344 . . . 4 (𝑥 = 𝐴 → ((((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦) ↔ (((𝑆𝐴) = 1 → (𝑆𝑦) = 1) → 𝐴𝑦)))
8 fveqeq2 6851 . . . . . 6 (𝑦 = 𝐵 → ((𝑆𝑦) = 1 ↔ (𝑆𝐵) = 1))
98imbi2d 340 . . . . 5 (𝑦 = 𝐵 → (((𝑆𝐴) = 1 → (𝑆𝑦) = 1) ↔ ((𝑆𝐴) = 1 → (𝑆𝐵) = 1)))
10 sseq2 3970 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
119, 10imbi12d 344 . . . 4 (𝑦 = 𝐵 → ((((𝑆𝐴) = 1 → (𝑆𝑦) = 1) → 𝐴𝑦) ↔ (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵)))
127, 11rspc2v 3590 . . 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 505 1 (𝜑 → (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wss 3910  cfv 6496  1c1 11051   C cch 29869  Statescst 29902
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504
This theorem is referenced by:  stcltr2i  31215  stcltrlem2  31217
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