Theorem cgrrflx2d

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

	Ref	Expression
Hypotheses	cgrrflx2d.1	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
	cgrrflx2d.2	⊢ ( 𝜑 → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
	cgrrflx2d.3	⊢ ( 𝜑 → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
Assertion	cgrrflx2d	⊢ ( 𝜑 → ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐵 , 𝐴 ⟩ )

Proof

Step	Hyp	Ref	Expression
1		cgrrflx2d.1	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
2		cgrrflx2d.2	⊢ ( 𝜑 → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
3		cgrrflx2d.3	⊢ ( 𝜑 → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
4		axcgrrflx	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐵 , 𝐴 ⟩ )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐵 , 𝐴 ⟩ )