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 )