Theorem cbvoprab1davw

Description: Change the first bound variable in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypothesis	cbvoprab1davw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑤 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	cbvoprab1davw	⊢ ( 𝜑 → { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } = { ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜒 } )

Proof

Step	Hyp	Ref	Expression
1		cbvoprab1davw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑤 ) → ( 𝜓 ↔ 𝜒 ) )
2		opeq1	⊢ ( 𝑥 = 𝑤 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑤 , 𝑦 ⟩ )
3	2	adantl	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑤 ) → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑤 , 𝑦 ⟩ )
4	3	opeq1d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑤 ) → ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ = ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑧 ⟩ )
5	4	eqeq2d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑤 ) → ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ↔ 𝑡 = ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑧 ⟩ ) )
6	5 1	anbi12d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑤 ) → ( ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ↔ ( 𝑡 = ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) ) )
7	6	2exbidv	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑤 ) → ( ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) ) )
8	7	cbvexdvaw	⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑤 ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) ) )
9	8	abbidv	⊢ ( 𝜑 → { 𝑡 ∣ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) } = { 𝑡 ∣ ∃ 𝑤 ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) } )
10		df-oprab	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } = { 𝑡 ∣ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) }
11		df-oprab	⊢ { ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜒 } = { 𝑡 ∣ ∃ 𝑤 ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) }
12	9 10 11	3eqtr4g	⊢ ( 𝜑 → { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } = { ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜒 } )