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Metamath Proof Explorer

Theorem 3ecoptocl

Description: Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995)

Ref Expression
Hypotheses 3ecoptocl.1 ⊢ 𝑆 = ( ( 𝐷 × 𝐷 ) / 𝑅 )
3ecoptocl.2 ⊢ ( [ ⟨ 𝑥 , 𝑦 ⟩ ] 𝑅 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3ecoptocl.3 ⊢ ( [ ⟨ 𝑧 , 𝑤 ⟩ ] 𝑅 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
3ecoptocl.4 ⊢ ( [ ⟨ 𝑣 , 𝑢 ⟩ ] 𝑅 = 𝐶 → ( 𝜒 ↔ 𝜃 ) )
3ecoptocl.5 ⊢ ( ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ) ∧ ( 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷 ) ∧ ( 𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐷 ) ) → 𝜑 )
Assertion 3ecoptocl ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → 𝜃 )

Proof

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