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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 → ( 𝜑 ↔ 𝜓 ) )
3optocl.3 ⊢ ( ⟨ 𝑧 , 𝑤 ⟩ = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
3optocl.4 ⊢ ( ⟨ 𝑣 , 𝑢 ⟩ = 𝐶 → ( 𝜒 ↔ 𝜃 ) )
3optocl.5 ⊢ ( ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐹 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐹 ) ∧ ( 𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹 ) ) → 𝜑 )
Assertion 3optocl ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑅 ) → 𝜃 )

Proof

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