# Metamath Proof Explorer

## Theorem 3optocl

Description: Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995)

Ref Expression
Hypotheses 3optocl.1 𝑅 = ( 𝐷 × 𝐹 )
3optocl.2 ( ⟨ 𝑥 , 𝑦 ⟩ = 𝐴 → ( 𝜑𝜓 ) )
3optocl.3 ( ⟨ 𝑧 , 𝑤 ⟩ = 𝐵 → ( 𝜓𝜒 ) )
3optocl.4 ( ⟨ 𝑣 , 𝑢 ⟩ = 𝐶 → ( 𝜒𝜃 ) )
3optocl.5 ( ( ( 𝑥𝐷𝑦𝐹 ) ∧ ( 𝑧𝐷𝑤𝐹 ) ∧ ( 𝑣𝐷𝑢𝐹 ) ) → 𝜑 )
Assertion 3optocl ( ( 𝐴𝑅𝐵𝑅𝐶𝑅 ) → 𝜃 )

### Proof

Step Hyp Ref Expression
1 3optocl.1 𝑅 = ( 𝐷 × 𝐹 )
2 3optocl.2 ( ⟨ 𝑥 , 𝑦 ⟩ = 𝐴 → ( 𝜑𝜓 ) )
3 3optocl.3 ( ⟨ 𝑧 , 𝑤 ⟩ = 𝐵 → ( 𝜓𝜒 ) )
4 3optocl.4 ( ⟨ 𝑣 , 𝑢 ⟩ = 𝐶 → ( 𝜒𝜃 ) )
5 3optocl.5 ( ( ( 𝑥𝐷𝑦𝐹 ) ∧ ( 𝑧𝐷𝑤𝐹 ) ∧ ( 𝑣𝐷𝑢𝐹 ) ) → 𝜑 )
6 4 imbi2d ( ⟨ 𝑣 , 𝑢 ⟩ = 𝐶 → ( ( ( 𝐴𝑅𝐵𝑅 ) → 𝜒 ) ↔ ( ( 𝐴𝑅𝐵𝑅 ) → 𝜃 ) ) )
7 2 imbi2d ( ⟨ 𝑥 , 𝑦 ⟩ = 𝐴 → ( ( ( 𝑣𝐷𝑢𝐹 ) → 𝜑 ) ↔ ( ( 𝑣𝐷𝑢𝐹 ) → 𝜓 ) ) )
8 3 imbi2d ( ⟨ 𝑧 , 𝑤 ⟩ = 𝐵 → ( ( ( 𝑣𝐷𝑢𝐹 ) → 𝜓 ) ↔ ( ( 𝑣𝐷𝑢𝐹 ) → 𝜒 ) ) )
9 5 3expia ( ( ( 𝑥𝐷𝑦𝐹 ) ∧ ( 𝑧𝐷𝑤𝐹 ) ) → ( ( 𝑣𝐷𝑢𝐹 ) → 𝜑 ) )
10 1 7 8 9 2optocl ( ( 𝐴𝑅𝐵𝑅 ) → ( ( 𝑣𝐷𝑢𝐹 ) → 𝜒 ) )
11 10 com12 ( ( 𝑣𝐷𝑢𝐹 ) → ( ( 𝐴𝑅𝐵𝑅 ) → 𝜒 ) )
12 1 6 11 optocl ( 𝐶𝑅 → ( ( 𝐴𝑅𝐵𝑅 ) → 𝜃 ) )
13 12 impcom ( ( ( 𝐴𝑅𝐵𝑅 ) ∧ 𝐶𝑅 ) → 𝜃 )
14 13 3impa ( ( 𝐴𝑅𝐵𝑅𝐶𝑅 ) → 𝜃 )