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Theorem dmdi 29495
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 29492 . . . . 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 3774 . . . . . 6 (𝑥 = 𝐶 → (𝐵𝑥𝐵𝐶))
4 ineq1 3956 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥𝐴) = (𝐶𝐴))
54oveq1d 6807 . . . . . . 7 (𝑥 = 𝐶 → ((𝑥𝐴) ∨ 𝐵) = ((𝐶𝐴) ∨ 𝐵))
6 ineq1 3956 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 ∩ (𝐴 𝐵)) = (𝐶 ∩ (𝐴 𝐵)))
75, 6eqeq12d 2785 . . . . . 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 3454 . . . 4 (𝐶C → (∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))) → (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
102, 9sylan9 491 . . 3 (((𝐴C𝐵C ) ∧ 𝐶C ) → (𝐴 𝑀* 𝐵 → (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
11103impa 1099 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝑀* 𝐵 → (𝐵𝐶 → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))))
1211imp32 405 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  cin 3720  wss 3721   class class class wbr 4784  (class class class)co 6792   C cch 28120   chj 28124   𝑀* cdmd 28158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-iota 5994  df-fv 6039  df-ov 6795  df-dmd 29474
This theorem is referenced by:  dmdi2  29497  dmdsl3  29508  csmdsymi  29527  mdsymlem1  29596
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