Theorem mdi 29679

Description: Consequence of the modular pair property. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
mdi	⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ 𝐵 ∧ 𝐶 ⊆ 𝐵)) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = (𝐶 ∨ℋ (𝐴 ∩ 𝐵)))

Proof of Theorem mdi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdbr 29678	. . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))))
2	1	biimpd 221	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 → ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))))
3		sseq1 3822	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑥 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐵))
4		oveq1 6885	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝑥 ∨ℋ 𝐴) = (𝐶 ∨ℋ 𝐴))
5	4	ineq1d 4011	. . . . . . 7 ⊢ (𝑥 = 𝐶 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = ((𝐶 ∨ℋ 𝐴) ∩ 𝐵))
6		oveq1 6885	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) = (𝐶 ∨ℋ (𝐴 ∩ 𝐵)))
7	5, 6	eqeq12d 2814	. . . . . 6 ⊢ (𝑥 = 𝐶 → (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = (𝐶 ∨ℋ (𝐴 ∩ 𝐵))))
8	3, 7	imbi12d 336	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))) ↔ (𝐶 ⊆ 𝐵 → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = (𝐶 ∨ℋ (𝐴 ∩ 𝐵)))))
9	8	rspcv 3493	. . . 4 ⊢ (𝐶 ∈ Cℋ → (∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))) → (𝐶 ⊆ 𝐵 → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = (𝐶 ∨ℋ (𝐴 ∩ 𝐵)))))
10	2, 9	sylan9 504	. . 3 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐶 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 → (𝐶 ⊆ 𝐵 → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = (𝐶 ∨ℋ (𝐴 ∩ 𝐵)))))
11	10	3impa 1137	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 → (𝐶 ⊆ 𝐵 → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = (𝐶 ∨ℋ (𝐴 ∩ 𝐵)))))
12	11	imp32 410	1 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ 𝐵 ∧ 𝐶 ⊆ 𝐵)) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = (𝐶 ∨ℋ (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 385   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157  ∀wral 3089   ∩ cin 3768   ⊆ wss 3769   class class class wbr 4843  (class class class)co 6878   C_ℋ cch 28311   ∨_ℋ chj 28315   𝑀_ℋ cmd 28348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-iota 6064  df-fv 6109  df-ov 6881  df-md 29664

This theorem is referenced by:  mdsl3  29700  mdslmd3i  29716  mdexchi  29719  atabsi  29785