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Theorem mdi 31535
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 31534 . . . . 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 4006 . . . . . 6 (𝑥 = 𝐶 → (𝑥𝐵𝐶𝐵))
4 oveq1 7412 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥 𝐴) = (𝐶 𝐴))
54ineq1d 4210 . . . . . . 7 (𝑥 = 𝐶 → ((𝑥 𝐴) ∩ 𝐵) = ((𝐶 𝐴) ∩ 𝐵))
6 oveq1 7412 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 (𝐴𝐵)) = (𝐶 (𝐴𝐵)))
75, 6eqeq12d 2748 . . . . . 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 3608 . . . 4 (𝐶C → (∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))) → (𝐶𝐵 → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))))
102, 9sylan9 508 . . 3 (((𝐴C𝐵C ) ∧ 𝐶C ) → (𝐴 𝑀 𝐵 → (𝐶𝐵 → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))))
11103impa 1110 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝑀 𝐵 → (𝐶𝐵 → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))))
1211imp32 419 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀 𝐵𝐶𝐵)) → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  cin 3946  wss 3947   class class class wbr 5147  (class class class)co 7405   C cch 30169   chj 30173   𝑀 cmd 30206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-iota 6492  df-fv 6548  df-ov 7408  df-md 31520
This theorem is referenced by:  mdsl3  31556  mdslmd3i  31572  mdexchi  31575  atabsi  31641
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