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Theorem dmdi 29686
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 29683 . . . . 5 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
21biimpd 221 . . . 4 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 → ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
3 sseq2 3823 . . . . . 6 (𝑥 = 𝐶 → (𝐵𝑥𝐵𝐶))
4 ineq1 4005 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥𝐴) = (𝐶𝐴))
54oveq1d 6893 . . . . . . 7 (𝑥 = 𝐶 → ((𝑥𝐴) ∨ 𝐵) = ((𝐶𝐴) ∨ 𝐵))
6 ineq1 4005 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 ∩ (𝐴 𝐵)) = (𝐶 ∩ (𝐴 𝐵)))
75, 6eqeq12d 2814 . . . . . 6 (𝑥 = 𝐶 → (((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)) ↔ ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵))))
83, 7imbi12d 336 . . . . 5 (𝑥 = 𝐶 → ((𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))) ↔ (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
98rspcv 3493 . . . 4 (𝐶C → (∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))) → (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
102, 9sylan9 504 . . 3 (((𝐴C𝐵C ) ∧ 𝐶C ) → (𝐴 𝑀* 𝐵 → (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
11103impa 1137 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝑀* 𝐵 → (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
1211imp32 410 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3089  cin 3768  wss 3769   class class class wbr 4843  (class class class)co 6878   C cch 28311   chj 28315   𝑀* cdmd 28349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-iota 6064  df-fv 6109  df-ov 6881  df-dmd 29665
This theorem is referenced by:  dmdi2  29688  dmdsl3  29699  csmdsymi  29718  mdsymlem1  29787
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