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Metamath Proof Explorer

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 )