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Theorem dmdi 32331
Description: Consequence of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
dmdi (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))

Proof of Theorem dmdi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dmdbr 32328 . . . . 5 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
21biimpd 229 . . . 4 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 → ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
3 sseq2 4022 . . . . . 6 (𝑥 = 𝐶 → (𝐵𝑥𝐵𝐶))
4 ineq1 4221 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥𝐴) = (𝐶𝐴))
54oveq1d 7446 . . . . . . 7 (𝑥 = 𝐶 → ((𝑥𝐴) ∨ 𝐵) = ((𝐶𝐴) ∨ 𝐵))
6 ineq1 4221 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 ∩ (𝐴 𝐵)) = (𝐶 ∩ (𝐴 𝐵)))
75, 6eqeq12d 2751 . . . . . 6 (𝑥 = 𝐶 → (((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)) ↔ ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵))))
83, 7imbi12d 344 . . . . 5 (𝑥 = 𝐶 → ((𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))) ↔ (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
98rspcv 3618 . . . 4 (𝐶C → (∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))) → (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
102, 9sylan9 507 . . 3 (((𝐴C𝐵C ) ∧ 𝐶C ) → (𝐴 𝑀* 𝐵 → (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
11103impa 1109 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝑀* 𝐵 → (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
1211imp32 418 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  cin 3962  wss 3963   class class class wbr 5148  (class class class)co 7431   C cch 30958   chj 30962   𝑀* cdmd 30996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-iota 6516  df-fv 6571  df-ov 7434  df-dmd 32310
This theorem is referenced by:  dmdi2  32333  dmdsl3  32344  csmdsymi  32363  mdsymlem1  32432
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