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Theorem mdi 29463
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 29462 . . . . 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 3767 . . . . . 6 (𝑥 = 𝐶 → (𝑥𝐵𝐶𝐵))
4 oveq1 6820 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥 𝐴) = (𝐶 𝐴))
54ineq1d 3956 . . . . . . 7 (𝑥 = 𝐶 → ((𝑥 𝐴) ∩ 𝐵) = ((𝐶 𝐴) ∩ 𝐵))
6 oveq1 6820 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 (𝐴𝐵)) = (𝐶 (𝐴𝐵)))
75, 6eqeq12d 2775 . . . . . 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 3445 . . . 4 (𝐶C → (∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))) → (𝐶𝐵 → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))))
102, 9sylan9 692 . . 3 (((𝐴C𝐵C ) ∧ 𝐶C ) → (𝐴 𝑀 𝐵 → (𝐶𝐵 → ((𝐶 𝐴) ∩ 𝐵) = (𝐶 (𝐴𝐵)))))
11103impa 1101 . 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 1072   = wceq 1632  wcel 2139  wral 3050  cin 3714  wss 3715   class class class wbr 4804  (class class class)co 6813   C cch 28095   chj 28099   𝑀 cmd 28132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-iota 6012  df-fv 6057  df-ov 6816  df-md 29448
This theorem is referenced by:  mdsl3  29484  mdslmd3i  29500  mdexchi  29503  atabsi  29569
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