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Theorem stcltr1i 29463
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 fveq2 6353 . . . . . . 7 (𝑥 = 𝐴 → (𝑆𝑥) = (𝑆𝐴))
54eqeq1d 2762 . . . . . 6 (𝑥 = 𝐴 → ((𝑆𝑥) = 1 ↔ (𝑆𝐴) = 1))
65imbi1d 330 . . . . 5 (𝑥 = 𝐴 → (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) ↔ ((𝑆𝐴) = 1 → (𝑆𝑦) = 1)))
7 sseq1 3767 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
86, 7imbi12d 333 . . . 4 (𝑥 = 𝐴 → ((((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦) ↔ (((𝑆𝐴) = 1 → (𝑆𝑦) = 1) → 𝐴𝑦)))
9 fveq2 6353 . . . . . . 7 (𝑦 = 𝐵 → (𝑆𝑦) = (𝑆𝐵))
109eqeq1d 2762 . . . . . 6 (𝑦 = 𝐵 → ((𝑆𝑦) = 1 ↔ (𝑆𝐵) = 1))
1110imbi2d 329 . . . . 5 (𝑦 = 𝐵 → (((𝑆𝐴) = 1 → (𝑆𝑦) = 1) ↔ ((𝑆𝐴) = 1 → (𝑆𝐵) = 1)))
12 sseq2 3768 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
1311, 12imbi12d 333 . . . 4 (𝑦 = 𝐵 → ((((𝑆𝐴) = 1 → (𝑆𝑦) = 1) → 𝐴𝑦) ↔ (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵)))
148, 13rspc2v 3461 . . 3 ((𝐴C𝐵C ) → (∀𝑥C𝑦C (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦) → (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵)))
152, 3, 14mp2an 710 . 2 (∀𝑥C𝑦C (((𝑆𝑥) = 1 → (𝑆𝑦) = 1) → 𝑥𝑦) → (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵))
161, 15simplbiim 661 1 (𝜑 → (((𝑆𝐴) = 1 → (𝑆𝐵) = 1) → 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wss 3715  cfv 6049  1c1 10149   C cch 28116  Statescst 28149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057
This theorem is referenced by:  stcltr2i  29464  stcltrlem2  29466
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