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Theorem mdi 32323
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 32322 . . . . 5 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
21biimpd 229 . . . 4 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 → ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
3 sseq1 4020 . . . . . 6 (𝑥 = 𝐶 → (𝑥𝐵𝐶𝐵))
4 oveq1 7437 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥 𝐴) = (𝐶 𝐴))
54ineq1d 4226 . . . . . . 7 (𝑥 = 𝐶 → ((𝑥 𝐴) ∩ 𝐵) = ((𝐶 𝐴) ∩ 𝐵))
6 oveq1 7437 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 (𝐴𝐵)) = (𝐶 (𝐴𝐵)))
75, 6eqeq12d 2750 . . . . . 6 (𝑥 = 𝐶 → (((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)) ↔ ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵))))
83, 7imbi12d 344 . . . . 5 (𝑥 = 𝐶 → ((𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))) ↔ (𝐶𝐵 → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))))
98rspcv 3617 . . . 4 (𝐶C → (∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))) → (𝐶𝐵 → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))))
102, 9sylan9 507 . . 3 (((𝐴C𝐵C ) ∧ 𝐶C ) → (𝐴 𝑀 𝐵 → (𝐶𝐵 → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))))
11103impa 1109 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝑀 𝐵 → (𝐶𝐵 → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))))
1211imp32 418 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀 𝐵𝐶𝐵)) → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  cin 3961  wss 3962   class class class wbr 5147  (class class class)co 7430   C cch 30957   chj 30961   𝑀 cmd 30994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-iota 6515  df-fv 6570  df-ov 7433  df-md 32308
This theorem is referenced by:  mdsl3  32344  mdslmd3i  32360  mdexchi  32363  atabsi  32429
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