Theorem endofsegidand

Description: Deduction form of endofsegid . (Contributed by Scott Fenton, 15-Oct-2013) (Revised by Mario Carneiro, 19-Apr-2014)

	Ref	Expression
Hypotheses	endofsegidand.1	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
	endofsegidand.2	⊢ ( 𝜑 → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
	endofsegidand.3	⊢ ( 𝜑 → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
	endofsegidand.4	⊢ ( 𝜑 → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
	endofsegidand.5	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ )
	endofsegidand.6	⊢ ( ( 𝜑 ∧ 𝜓 ) → ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐴 , 𝐶 ⟩ )
Assertion	endofsegidand	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐵 = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		endofsegidand.1	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
2		endofsegidand.2	⊢ ( 𝜑 → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
3		endofsegidand.3	⊢ ( 𝜑 → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
4		endofsegidand.4	⊢ ( 𝜑 → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
5		endofsegidand.5	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ )
6		endofsegidand.6	⊢ ( ( 𝜑 ∧ 𝜓 ) → ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐴 , 𝐶 ⟩ )
7		endofsegid	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐴 , 𝐶 ⟩ ) → 𝐵 = 𝐶 ) )
8	1 2 4 3 7	syl13anc	⊢ ( 𝜑 → ( ( 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐴 , 𝐶 ⟩ ) → 𝐵 = 𝐶 ) )
9	8	adantr	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐴 , 𝐶 ⟩ ) → 𝐵 = 𝐶 ) )
10	5 6 9	mp2and	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐵 = 𝐶 )