Theorem cbvoprab12v

Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004)

	Ref	Expression
Hypothesis	cbvoprab12v.1	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑦 = 𝑣 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvoprab12v	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { ⟨ ⟨ 𝑤 , 𝑣 ⟩ , 𝑧 ⟩ ∣ 𝜓 }

Proof

Step	Hyp	Ref	Expression
1		cbvoprab12v.1	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑦 = 𝑣 ) → ( 𝜑 ↔ 𝜓 ) )
2		opeq12	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑦 = 𝑣 ) → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ )
3	2	opeq1d	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑦 = 𝑣 ) → ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ = ⟨ ⟨ 𝑤 , 𝑣 ⟩ , 𝑧 ⟩ )
4	3	eqeq2d	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑦 = 𝑣 ) → ( 𝑢 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ↔ 𝑢 = ⟨ ⟨ 𝑤 , 𝑣 ⟩ , 𝑧 ⟩ ) )
5	4 1	anbi12d	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑦 = 𝑣 ) → ( ( 𝑢 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜑 ) ↔ ( 𝑢 = ⟨ ⟨ 𝑤 , 𝑣 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ) )
6	5	exbidv	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑦 = 𝑣 ) → ( ∃ 𝑧 ( 𝑢 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑧 ( 𝑢 = ⟨ ⟨ 𝑤 , 𝑣 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ) )
7	6	cbvex2vw	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑢 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑤 ∃ 𝑣 ∃ 𝑧 ( 𝑢 = ⟨ ⟨ 𝑤 , 𝑣 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) )
8	7	abbii	⊢ { 𝑢 ∣ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑢 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜑 ) } = { 𝑢 ∣ ∃ 𝑤 ∃ 𝑣 ∃ 𝑧 ( 𝑢 = ⟨ ⟨ 𝑤 , 𝑣 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) }
9		df-oprab	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { 𝑢 ∣ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑢 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜑 ) }
10		df-oprab	⊢ { ⟨ ⟨ 𝑤 , 𝑣 ⟩ , 𝑧 ⟩ ∣ 𝜓 } = { 𝑢 ∣ ∃ 𝑤 ∃ 𝑣 ∃ 𝑧 ( 𝑢 = ⟨ ⟨ 𝑤 , 𝑣 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) }
11	8 9 10	3eqtr4i	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { ⟨ ⟨ 𝑤 , 𝑣 ⟩ , 𝑧 ⟩ ∣ 𝜓 }