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Metamath Proof Explorer

Theorem 2optocl

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

Ref Expression
Hypotheses 2optocl.1 ⊢ 𝑅 = ( 𝐶 × 𝐷 )
2optocl.2 ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2optocl.3 ⊢ ( ⟨ 𝑧 , 𝑤 ⟩ = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
2optocl.4 ⊢ ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ( 𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷 ) ) → 𝜑 )
Assertion 2optocl ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅 ) → 𝜒 )

Proof

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