Theorem cbvoprab13vw

Description: Change the first and third bound variables in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypothesis	cbvoprab13vw.1	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑧 = 𝑣 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	cbvoprab13vw	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } = { ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑣 ⟩ ∣ 𝜒 }

Proof

Step	Hyp	Ref	Expression
1		cbvoprab13vw.1	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑧 = 𝑣 ) → ( 𝜓 ↔ 𝜒 ) )
2		opeq1	⊢ ( 𝑥 = 𝑤 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑤 , 𝑦 ⟩ )
3	2	adantr	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑧 = 𝑣 ) → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑤 , 𝑦 ⟩ )
4		simpr	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑧 = 𝑣 ) → 𝑧 = 𝑣 )
5	3 4	opeq12d	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑧 = 𝑣 ) → ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ = ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑣 ⟩ )
6	5	eqeq2d	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑧 = 𝑣 ) → ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ↔ 𝑡 = ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑣 ⟩ ) )
7	6 1	anbi12d	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑧 = 𝑣 ) → ( ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ↔ ( 𝑡 = ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑣 ⟩ ∧ 𝜒 ) ) )
8	7	cbvexdvaw	⊢ ( 𝑥 = 𝑤 → ( ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑣 ( 𝑡 = ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑣 ⟩ ∧ 𝜒 ) ) )
9	8	exbidv	⊢ ( 𝑥 = 𝑤 → ( ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑦 ∃ 𝑣 ( 𝑡 = ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑣 ⟩ ∧ 𝜒 ) ) )
10	9	cbvexvw	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑤 ∃ 𝑦 ∃ 𝑣 ( 𝑡 = ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑣 ⟩ ∧ 𝜒 ) )
11	10	abbii	⊢ { 𝑡 ∣ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) } = { 𝑡 ∣ ∃ 𝑤 ∃ 𝑦 ∃ 𝑣 ( 𝑡 = ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑣 ⟩ ∧ 𝜒 ) }
12		df-oprab	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } = { 𝑡 ∣ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑡 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) }
13		df-oprab	⊢ { ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑣 ⟩ ∣ 𝜒 } = { 𝑡 ∣ ∃ 𝑤 ∃ 𝑦 ∃ 𝑣 ( 𝑡 = ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑣 ⟩ ∧ 𝜒 ) }
14	11 12 13	3eqtr4i	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } = { ⟨ ⟨ 𝑤 , 𝑦 ⟩ , 𝑣 ⟩ ∣ 𝜒 }