Metamath Proof Explorer

Theorem btwnexchand

Description: Deduction form of btwnexch . (Contributed by Scott Fenton, 13-Oct-2013)

Ref Expression
Hypotheses btwnexchand.1 ⊢ φ → N ∈ ℕ
btwnexchand.2 ⊢ φ → A ∈ 𝔼 ⁡ N
btwnexchand.3 ⊢ φ → B ∈ 𝔼 ⁡ N
btwnexchand.4 ⊢ φ → C ∈ 𝔼 ⁡ N
btwnexchand.5 ⊢ φ → D ∈ 𝔼 ⁡ N
btwnexchand.6 ⊢ φ ∧ ψ → B Btwn A C
btwnexchand.7 ⊢ φ ∧ ψ → C Btwn A D
Assertion btwnexchand ⊢ φ ∧ ψ → B Btwn A D

Proof

Step Hyp Ref Expression
1 btwnexchand.1 ⊢ φ → N ∈ ℕ
2 btwnexchand.2 ⊢ φ → A ∈ 𝔼 ⁡ N
3 btwnexchand.3 ⊢ φ → B ∈ 𝔼 ⁡ N
4 btwnexchand.4 ⊢ φ → C ∈ 𝔼 ⁡ N
5 btwnexchand.5 ⊢ φ → D ∈ 𝔼 ⁡ N
6 btwnexchand.6 ⊢ φ ∧ ψ → B Btwn A C
7 btwnexchand.7 ⊢ φ ∧ ψ → C Btwn A D
8 btwnexch ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → B Btwn A C ∧ C Btwn A D → B Btwn A D
9 1 2 3 4 5 8 syl122anc ⊢ φ → B Btwn A C ∧ C Btwn A D → B Btwn A D
10 9 adantr ⊢ φ ∧ ψ → B Btwn A C ∧ C Btwn A D → B Btwn A D
11 6 7 10 mp2and ⊢ φ ∧ ψ → B Btwn A D