Theorem cbvopabdavw

Description: Change bound variables in an ordered-pair class abstraction. Deduction form. (Contributed by GG, 14-Aug-2025)

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

Proof

Step	Hyp	Ref	Expression
1		cbvopabdavw.1	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → ( 𝜓 ↔ 𝜒 ) )
2		simplr	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → 𝑥 = 𝑧 )
3		simpr	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → 𝑦 = 𝑤 )
4	2 3	opeq12d	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑧 , 𝑤 ⟩ )
5	4	eqeq2d	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → ( 𝑡 = ⟨ 𝑥 , 𝑦 ⟩ ↔ 𝑡 = ⟨ 𝑧 , 𝑤 ⟩ ) )
6	5 1	anbi12d	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → ( ( 𝑡 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ ( 𝑡 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝜒 ) ) )
7	6	cbvexdvaw	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑧 ) → ( ∃ 𝑦 ( 𝑡 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑤 ( 𝑡 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝜒 ) ) )
8	7	cbvexdvaw	⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ( 𝑡 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑧 ∃ 𝑤 ( 𝑡 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝜒 ) ) )
9	8	abbidv	⊢ ( 𝜑 → { 𝑡 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑡 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) } = { 𝑡 ∣ ∃ 𝑧 ∃ 𝑤 ( 𝑡 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝜒 ) } )
10		df-opab	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } = { 𝑡 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑡 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) }
11		df-opab	⊢ { ⟨ 𝑧 , 𝑤 ⟩ ∣ 𝜒 } = { 𝑡 ∣ ∃ 𝑧 ∃ 𝑤 ( 𝑡 = ⟨ 𝑧 , 𝑤 ⟩ ∧ 𝜒 ) }
12	9 10 11	3eqtr4g	⊢ ( 𝜑 → { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } = { ⟨ 𝑧 , 𝑤 ⟩ ∣ 𝜒 } )