Theorem cgrrflxd

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

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

Proof

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