Theorem cbvoprab2vw

Description: Change the second bound variable in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypothesis	cbvoprab2vw.1	⊢ ( 𝑦 = 𝑤 → ( 𝜓 ↔ 𝜒 ) )
Assertion	cbvoprab2vw	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } = { ⟨ ⟨ 𝑥 , 𝑤 ⟩ , 𝑧 ⟩ ∣ 𝜒 }

Proof

Step	Hyp	Ref	Expression
1		cbvoprab2vw.1	⊢ ( 𝑦 = 𝑤 → ( 𝜓 ↔ 𝜒 ) )
2		opeq2	⊢ ( 𝑦 = 𝑤 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑥 , 𝑤 ⟩ )
3	2	opeq1d	⊢ ( 𝑦 = 𝑤 → ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ = ⟨ ⟨ 𝑥 , 𝑤 ⟩ , 𝑧 ⟩ )
4	3	eqeq2d	⊢ ( 𝑦 = 𝑤 → ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ↔ 𝑡 = ⟨ ⟨ 𝑥 , 𝑤 ⟩ , 𝑧 ⟩ ) )
5	4 1	anbi12d	⊢ ( 𝑦 = 𝑤 → ( ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ↔ ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑤 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) ) )
6	5	exbidv	⊢ ( 𝑦 = 𝑤 → ( ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑤 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) ) )
7	6	cbvexvw	⊢ ( ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑤 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑤 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) )
8	7	exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑥 ∃ 𝑤 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑤 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) )
9	8	abbii	⊢ { 𝑡 ∣ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) } = { 𝑡 ∣ ∃ 𝑥 ∃ 𝑤 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑤 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) }
10		df-oprab	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } = { 𝑡 ∣ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) }
11		df-oprab	⊢ { ⟨ ⟨ 𝑥 , 𝑤 ⟩ , 𝑧 ⟩ ∣ 𝜒 } = { 𝑡 ∣ ∃ 𝑥 ∃ 𝑤 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑤 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) }
12	9 10 11	3eqtr4i	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } = { ⟨ ⟨ 𝑥 , 𝑤 ⟩ , 𝑧 ⟩ ∣ 𝜒 }