Theorem cbvopab2

Description: Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013)

	Ref	Expression
Hypotheses	cbvopab2.1	⊢ Ⅎ 𝑧 𝜑
	cbvopab2.2	⊢ Ⅎ 𝑦 𝜓
	cbvopab2.3	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvopab2	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑥 , 𝑧 ⟩ ∣ 𝜓 }

Proof

Step	Hyp	Ref	Expression
1		cbvopab2.1	⊢ Ⅎ 𝑧 𝜑
2		cbvopab2.2	⊢ Ⅎ 𝑦 𝜓
3		cbvopab2.3	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
4		nfv	⊢ Ⅎ 𝑧 𝑤 = ⟨ 𝑥 , 𝑦 ⟩
5	4 1	nfan	⊢ Ⅎ 𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 )
6		nfv	⊢ Ⅎ 𝑦 𝑤 = ⟨ 𝑥 , 𝑧 ⟩
7	6 2	nfan	⊢ Ⅎ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 )
8		opeq2	⊢ ( 𝑦 = 𝑧 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑥 , 𝑧 ⟩ )
9	8	eqeq2d	⊢ ( 𝑦 = 𝑧 → ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ↔ 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ) )
10	9 3	anbi12d	⊢ ( 𝑦 = 𝑧 → ( ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 ) ) )
11	5 7 10	cbvexv1	⊢ ( ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 ) )
12	11	exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 ) )
13	12	abbii	⊢ { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 ) }
14		df-opab	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) }
15		df-opab	⊢ { ⟨ 𝑥 , 𝑧 ⟩ ∣ 𝜓 } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 ) }
16	13 14 15	3eqtr4i	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑥 , 𝑧 ⟩ ∣ 𝜓 }