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Theorem dmdi 29391
 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 29388 . . . . 5 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
21biimpd 219 . . . 4 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 → ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
3 sseq2 3733 . . . . . 6 (𝑥 = 𝐶 → (𝐵𝑥𝐵𝐶))
4 ineq1 3915 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥𝐴) = (𝐶𝐴))
54oveq1d 6780 . . . . . . 7 (𝑥 = 𝐶 → ((𝑥𝐴) ∨ 𝐵) = ((𝐶𝐴) ∨ 𝐵))
6 ineq1 3915 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 ∩ (𝐴 𝐵)) = (𝐶 ∩ (𝐴 𝐵)))
75, 6eqeq12d 2739 . . . . . 6 (𝑥 = 𝐶 → (((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)) ↔ ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵))))
83, 7imbi12d 333 . . . . 5 (𝑥 = 𝐶 → ((𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))) ↔ (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
98rspcv 3409 . . . 4 (𝐶C → (∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))) → (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
102, 9sylan9 692 . . 3 (((𝐴C𝐵C ) ∧ 𝐶C ) → (𝐴 𝑀* 𝐵 → (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
11103impa 1100 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝑀* 𝐵 → (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
1211imp32 448 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1596   ∈ wcel 2103  ∀wral 3014   ∩ cin 3679   ⊆ wss 3680   class class class wbr 4760  (class class class)co 6765   Cℋ cch 28016   ∨ℋ chj 28020   𝑀ℋ* cdmd 28054 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-iota 5964  df-fv 6009  df-ov 6768  df-dmd 29370 This theorem is referenced by:  dmdi2  29393  dmdsl3  29404  csmdsymi  29423  mdsymlem1  29492
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