Theorem cbvoprab12

Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004) (Proof shortened by Andrew Salmon, 22-Oct-2011)

	Ref	Expression
Hypotheses	cbvoprab12.1	⊢ Ⅎ 𝑤 𝜑
	cbvoprab12.2	⊢ Ⅎ 𝑣 𝜑
	cbvoprab12.3	⊢ Ⅎ 𝑥 𝜓
	cbvoprab12.4	⊢ Ⅎ 𝑦 𝜓
	cbvoprab12.5	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑦 = 𝑣 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvoprab12	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { ⟨ ⟨ 𝑤 , 𝑣 ⟩ , 𝑧 ⟩ ∣ 𝜓 }

Proof

Step	Hyp	Ref	Expression
1		cbvoprab12.1	⊢ Ⅎ 𝑤 𝜑
2		cbvoprab12.2	⊢ Ⅎ 𝑣 𝜑
3		cbvoprab12.3	⊢ Ⅎ 𝑥 𝜓
4		cbvoprab12.4	⊢ Ⅎ 𝑦 𝜓
5		cbvoprab12.5	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑦 = 𝑣 ) → ( 𝜑 ↔ 𝜓 ) )
6		nfv	⊢ Ⅎ 𝑤 𝑢 = ⟨ 𝑥 , 𝑦 ⟩
7	6 1	nfan	⊢ Ⅎ 𝑤 ( 𝑢 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 )
8		nfv	⊢ Ⅎ 𝑣 𝑢 = ⟨ 𝑥 , 𝑦 ⟩
9	8 2	nfan	⊢ Ⅎ 𝑣 ( 𝑢 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 )
10		nfv	⊢ Ⅎ 𝑥 𝑢 = ⟨ 𝑤 , 𝑣 ⟩
11	10 3	nfan	⊢ Ⅎ 𝑥 ( 𝑢 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝜓 )
12		nfv	⊢ Ⅎ 𝑦 𝑢 = ⟨ 𝑤 , 𝑣 ⟩
13	12 4	nfan	⊢ Ⅎ 𝑦 ( 𝑢 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝜓 )
14		opeq12	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑦 = 𝑣 ) → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑤 , 𝑣 ⟩ )
15	14	eqeq2d	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑦 = 𝑣 ) → ( 𝑢 = ⟨ 𝑥 , 𝑦 ⟩ ↔ 𝑢 = ⟨ 𝑤 , 𝑣 ⟩ ) )
16	15 5	anbi12d	⊢ ( ( 𝑥 = 𝑤 ∧ 𝑦 = 𝑣 ) → ( ( 𝑢 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ( 𝑢 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝜓 ) ) )
17	7 9 11 13 16	cbvex2v	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑢 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑤 ∃ 𝑣 ( 𝑢 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝜓 ) )
18	17	opabbii	⊢ { ⟨ 𝑢 , 𝑧 ⟩ ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑢 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) } = { ⟨ 𝑢 , 𝑧 ⟩ ∣ ∃ 𝑤 ∃ 𝑣 ( 𝑢 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝜓 ) }
19		dfoprab2	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { ⟨ 𝑢 , 𝑧 ⟩ ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑢 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) }
20		dfoprab2	⊢ { ⟨ ⟨ 𝑤 , 𝑣 ⟩ , 𝑧 ⟩ ∣ 𝜓 } = { ⟨ 𝑢 , 𝑧 ⟩ ∣ ∃ 𝑤 ∃ 𝑣 ( 𝑢 = ⟨ 𝑤 , 𝑣 ⟩ ∧ 𝜓 ) }
21	18 19 20	3eqtr4i	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { ⟨ ⟨ 𝑤 , 𝑣 ⟩ , 𝑧 ⟩ ∣ 𝜓 }