Metamath Proof Explorer


Theorem cgrcomland

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

Ref Expression
Hypotheses cgrcomlrand.1 ( 𝜑𝑁 ∈ ℕ )
cgrcomlrand.2 ( 𝜑𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
cgrcomlrand.3 ( 𝜑𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
cgrcomlrand.4 ( 𝜑𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
cgrcomlrand.5 ( 𝜑𝐷 ∈ ( 𝔼 ‘ 𝑁 ) )
cgrcomlrand.6 ( ( 𝜑𝜓 ) → ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
Assertion cgrcomland ( ( 𝜑𝜓 ) → ⟨ 𝐵 , 𝐴 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )

Proof

Step Hyp Ref Expression
1 cgrcomlrand.1 ( 𝜑𝑁 ∈ ℕ )
2 cgrcomlrand.2 ( 𝜑𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
3 cgrcomlrand.3 ( 𝜑𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
4 cgrcomlrand.4 ( 𝜑𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
5 cgrcomlrand.5 ( 𝜑𝐷 ∈ ( 𝔼 ‘ 𝑁 ) )
6 cgrcomlrand.6 ( ( 𝜑𝜓 ) → ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
7 cgrcoml ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ↔ ⟨ 𝐵 , 𝐴 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) )
8 1 2 3 4 5 7 syl122anc ( 𝜑 → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ↔ ⟨ 𝐵 , 𝐴 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) )
9 8 adantr ( ( 𝜑𝜓 ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ↔ ⟨ 𝐵 , 𝐴 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) )
10 6 9 mpbid ( ( 𝜑𝜓 ) → ⟨ 𝐵 , 𝐴 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )