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Theorem mdi 29282
Description: Consequence of the modular pair property. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
mdi (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀 𝐵𝐶𝐵)) → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))

Proof of Theorem mdi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 mdbr 29281 . . . . 5 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
21biimpd 219 . . . 4 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 → ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
3 sseq1 3659 . . . . . 6 (𝑥 = 𝐶 → (𝑥𝐵𝐶𝐵))
4 oveq1 6697 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥 𝐴) = (𝐶 𝐴))
54ineq1d 3846 . . . . . . 7 (𝑥 = 𝐶 → ((𝑥 𝐴) ∩ 𝐵) = ((𝐶 𝐴) ∩ 𝐵))
6 oveq1 6697 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 (𝐴𝐵)) = (𝐶 (𝐴𝐵)))
75, 6eqeq12d 2666 . . . . . 6 (𝑥 = 𝐶 → (((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)) ↔ ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵))))
83, 7imbi12d 333 . . . . 5 (𝑥 = 𝐶 → ((𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))) ↔ (𝐶𝐵 → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))))
98rspcv 3336 . . . 4 (𝐶C → (∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))) → (𝐶𝐵 → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))))
102, 9sylan9 690 . . 3 (((𝐴C𝐵C ) ∧ 𝐶C ) → (𝐴 𝑀 𝐵 → (𝐶𝐵 → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))))
11103impa 1278 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝑀 𝐵 → (𝐶𝐵 → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))))
1211imp32 448 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀 𝐵𝐶𝐵)) → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cin 3606  wss 3607   class class class wbr 4685  (class class class)co 6690   C cch 27914   chj 27918   𝑀 cmd 27951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-iota 5889  df-fv 5934  df-ov 6693  df-md 29267
This theorem is referenced by:  mdsl3  29303  mdslmd3i  29319  mdexchi  29322  atabsi  29388
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