Theorem btwncomand

Description: Deduction form of btwncom . (Contributed by Scott Fenton, 14-Oct-2013)

	Ref	Expression
Hypotheses	btwncomand.1	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
	btwncomand.2	⊢ ( 𝜑 → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
	btwncomand.3	⊢ ( 𝜑 → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
	btwncomand.4	⊢ ( 𝜑 → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
	btwncomand.5	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ )
Assertion	btwncomand	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 Btwn ⟨ 𝐶 , 𝐵 ⟩ )

Proof

Step	Hyp	Ref	Expression
1		btwncomand.1	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
2		btwncomand.2	⊢ ( 𝜑 → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
3		btwncomand.3	⊢ ( 𝜑 → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
4		btwncomand.4	⊢ ( 𝜑 → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
5		btwncomand.5	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ )
6		btwncom	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ↔ 𝐴 Btwn ⟨ 𝐶 , 𝐵 ⟩ ) )
7	1 2 3 4 6	syl13anc	⊢ ( 𝜑 → ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ↔ 𝐴 Btwn ⟨ 𝐶 , 𝐵 ⟩ ) )
8	7	adantr	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ↔ 𝐴 Btwn ⟨ 𝐶 , 𝐵 ⟩ ) )
9	5 8	mpbid	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 Btwn ⟨ 𝐶 , 𝐵 ⟩ )