Theorem mdi 32327

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 32326	. . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))))
2	1	biimpd 229	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 → ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))))
3		sseq1 4034	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑥 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐵))
4		oveq1 7455	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝑥 ∨ℋ 𝐴) = (𝐶 ∨ℋ 𝐴))
5	4	ineq1d 4240	. . . . . . 7 ⊢ (𝑥 = 𝐶 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = ((𝐶 ∨ℋ 𝐴) ∩ 𝐵))
6		oveq1 7455	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) = (𝐶 ∨ℋ (𝐴 ∩ 𝐵)))
7	5, 6	eqeq12d 2756	. . . . . 6 ⊢ (𝑥 = 𝐶 → (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = (𝐶 ∨ℋ (𝐴 ∩ 𝐵))))
8	3, 7	imbi12d 344	. . . . 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 3631	. . . 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 507	. . 3 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐶 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 → (𝐶 ⊆ 𝐵 → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = (𝐶 ∨ℋ (𝐴 ∩ 𝐵)))))
11	10	3impa 1110	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 → (𝐶 ⊆ 𝐵 → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = (𝐶 ∨ℋ (𝐴 ∩ 𝐵)))))
12	11	imp32 418	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 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∩ cin 3975   ⊆ wss 3976   class class class wbr 5166  (class class class)co 7448   C_ℋ cch 30961   ∨_ℋ chj 30965   𝑀_ℋ cmd 30998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-iota 6525  df-fv 6581  df-ov 7451  df-md 32312

This theorem is referenced by:  mdsl3  32348  mdslmd3i  32364  mdexchi  32367  atabsi  32433