Theorem mdi

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

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

Proof

Step	Hyp	Ref	Expression
1		mdbr	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ 𝐵 ↔ ∀ 𝑥 ∈ Cℋ ( 𝑥 ⊆ 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) )
2	1	biimpd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ 𝐵 → ∀ 𝑥 ∈ Cℋ ( 𝑥 ⊆ 𝐵 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) )
3		sseq1	⊢ ( 𝑥 = 𝐶 → ( 𝑥 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐵 ) )
4		oveq1	⊢ ( 𝑥 = 𝐶 → ( 𝑥 ∨ℋ 𝐴 ) = ( 𝐶 ∨ℋ 𝐴 ) )
5	4	ineq1d	⊢ ( 𝑥 = 𝐶 → ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( ( 𝐶 ∨ℋ 𝐴 ) ∩ 𝐵 ) )
6		oveq1	⊢ ( 𝑥 = 𝐶 → ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐶 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) )
7	5 6	eqeq12d	⊢ ( 𝑥 = 𝐶 → ( ( ( 𝑥 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝑥 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ↔ ( ( 𝐶 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝐶 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) )
8	3 7	imbi12d	⊢ ( 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	⊢ ( 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	⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐶 ∈ Cℋ ) → ( 𝐴 𝑀ℋ 𝐵 → ( 𝐶 ⊆ 𝐵 → ( ( 𝐶 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝐶 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) )
11	10	3impa	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐴 𝑀ℋ 𝐵 → ( 𝐶 ⊆ 𝐵 → ( ( 𝐶 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝐶 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) ) ) )
12	11	imp32	⊢ ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ ( 𝐴 𝑀ℋ 𝐵 ∧ 𝐶 ⊆ 𝐵 ) ) → ( ( 𝐶 ∨ℋ 𝐴 ) ∩ 𝐵 ) = ( 𝐶 ∨ℋ ( 𝐴 ∩ 𝐵 ) ) )