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Metamath Proof Explorer

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 ) ) )