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 ⟩ )