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Metamath Proof Explorer

Theorem cbvopab2v

Description: Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999)

Ref Expression
Hypothesis cbvopab2v.1 ⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
Assertion cbvopab2v { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑥 , 𝑧 ⟩ ∣ 𝜓 }

Proof

Step Hyp Ref Expression
1 cbvopab2v.1 ⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
2 opeq2 ⊢ ( 𝑦 = 𝑧 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑥 , 𝑧 ⟩ )
3 2 eqeq2d ⊢ ( 𝑦 = 𝑧 → ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ↔ 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ) )
4 3 1 anbi12d ⊢ ( 𝑦 = 𝑧 → ( ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 ) ) )
5 4 cbvexvw ⊢ ( ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 ) )
6 5 exbii ⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 ) )
7 6 abbii ⊢ { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 ) }
8 df-opab ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) }
9 df-opab ⊢ { ⟨ 𝑥 , 𝑧 ⟩ ∣ 𝜓 } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 ) }
10 7 8 9 3eqtr4i ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑥 , 𝑧 ⟩ ∣ 𝜓 }