Theorem 2ecoptocl

Description: Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995)

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

Proof

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