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Theorem mdi 32098
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 32097 . . . . 5 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
21biimpd 228 . . . 4 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 → ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
3 sseq1 4003 . . . . . 6 (𝑥 = 𝐶 → (𝑥𝐵𝐶𝐵))
4 oveq1 7421 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥 𝐴) = (𝐶 𝐴))
54ineq1d 4207 . . . . . . 7 (𝑥 = 𝐶 → ((𝑥 𝐴) ∩ 𝐵) = ((𝐶 𝐴) ∩ 𝐵))
6 oveq1 7421 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 (𝐴𝐵)) = (𝐶 (𝐴𝐵)))
75, 6eqeq12d 2743 . . . . . 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 3603 . . . 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 1108 . 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 1085   = wceq 1534  wcel 2099  wral 3056  cin 3943  wss 3944   class class class wbr 5142  (class class class)co 7414   C cch 30732   chj 30736   𝑀 cmd 30769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-iota 6494  df-fv 6550  df-ov 7417  df-md 32083
This theorem is referenced by:  mdsl3  32119  mdslmd3i  32135  mdexchi  32138  atabsi  32204
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