Theorem mdi 32017

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 32016	. . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))))
2	1	biimpd 228	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 → ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))))
3		sseq1 3999	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑥 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐵))
4		oveq1 7408	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝑥 ∨ℋ 𝐴) = (𝐶 ∨ℋ 𝐴))
5	4	ineq1d 4203	. . . . . . 7 ⊢ (𝑥 = 𝐶 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = ((𝐶 ∨ℋ 𝐴) ∩ 𝐵))
6		oveq1 7408	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) = (𝐶 ∨ℋ (𝐴 ∩ 𝐵)))
7	5, 6	eqeq12d 2740	. . . . . 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 3600	. . . 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 1107	. 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 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3053   ∩ cin 3939   ⊆ wss 3940   class class class wbr 5138  (class class class)co 7401   C_ℋ cch 30651   ∨_ℋ chj 30655   𝑀_ℋ cmd 30688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-iota 6485  df-fv 6541  df-ov 7404  df-md 32002

This theorem is referenced by:  mdsl3  32038  mdslmd3i  32054  mdexchi  32057  atabsi  32123