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Theorem dmdi 30006
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 30003 . . . . 5 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
21biimpd 230 . . . 4 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 → ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
3 sseq2 3990 . . . . . 6 (𝑥 = 𝐶 → (𝐵𝑥𝐵𝐶))
4 ineq1 4178 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥𝐴) = (𝐶𝐴))
54oveq1d 7160 . . . . . . 7 (𝑥 = 𝐶 → ((𝑥𝐴) ∨ 𝐵) = ((𝐶𝐴) ∨ 𝐵))
6 ineq1 4178 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 ∩ (𝐴 𝐵)) = (𝐶 ∩ (𝐴 𝐵)))
75, 6eqeq12d 2834 . . . . . 6 (𝑥 = 𝐶 → (((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)) ↔ ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵))))
83, 7imbi12d 346 . . . . 5 (𝑥 = 𝐶 → ((𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))) ↔ (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
98rspcv 3615 . . . 4 (𝐶C → (∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))) → (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
102, 9sylan9 508 . . 3 (((𝐴C𝐵C ) ∧ 𝐶C ) → (𝐴 𝑀* 𝐵 → (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
11103impa 1102 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝑀* 𝐵 → (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
1211imp32 419 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  cin 3932  wss 3933   class class class wbr 5057  (class class class)co 7145   C cch 28633   chj 28637   𝑀* cdmd 28671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-iota 6307  df-fv 6356  df-ov 7148  df-dmd 29985
This theorem is referenced by:  dmdi2  30008  dmdsl3  30019  csmdsymi  30038  mdsymlem1  30107
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