Theorem cbvopab1davw

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

	Ref	Expression
Hypothesis	cbvopab1davw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑧 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	cbvopab1davw	⊢ ( 𝜑 → { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } = { ⟨ 𝑧 , 𝑦 ⟩ ∣ 𝜒 } )

Proof

Step	Hyp	Ref	Expression
1		cbvopab1davw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑧 ) → ( 𝜓 ↔ 𝜒 ) )
2		opeq1	⊢ ( 𝑥 = 𝑧 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑧 , 𝑦 ⟩ )
3	2	adantl	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑧 ) → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑧 , 𝑦 ⟩ )
4	3	eqeq2d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑧 ) → ( 𝑡 = ⟨ 𝑥 , 𝑦 ⟩ ↔ 𝑡 = ⟨ 𝑧 , 𝑦 ⟩ ) )
5	4 1	anbi12d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑧 ) → ( ( 𝑡 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ ( 𝑡 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜒 ) ) )
6	5	exbidv	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑧 ) → ( ∃ 𝑦 ( 𝑡 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑦 ( 𝑡 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜒 ) ) )
7	6	cbvexdvaw	⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ( 𝑡 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑧 ∃ 𝑦 ( 𝑡 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜒 ) ) )
8	7	abbidv	⊢ ( 𝜑 → { 𝑡 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑡 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) } = { 𝑡 ∣ ∃ 𝑧 ∃ 𝑦 ( 𝑡 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜒 ) } )
9		df-opab	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } = { 𝑡 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑡 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) }
10		df-opab	⊢ { ⟨ 𝑧 , 𝑦 ⟩ ∣ 𝜒 } = { 𝑡 ∣ ∃ 𝑧 ∃ 𝑦 ( 𝑡 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜒 ) }
11	8 9 10	3eqtr4g	⊢ ( 𝜑 → { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } = { ⟨ 𝑧 , 𝑦 ⟩ ∣ 𝜒 } )